Computer X-ray microtomography (µXCT) represents a powerful tool for
investigating the physical properties of porous rocks. While calculated
porosities determined by this method typically match experimental
measurements, computed permeabilities are often overestimated by more than
1 order of magnitude. This effect increases towards smaller pore sizes, as
shown in this study, in which nanostructural features related to clay
minerals reduce the permeability of tight reservoir sandstone samples.
Focussed ion beam scanning
electron microscopy (FIB-SEM) tomography was applied to determine the permeability effects of
illites at the nanometre scale, and Navier–Stokes equations were applied to
calculate the permeability of these domains. With these data, microporous
domains (porous voxels) were defined using microtomography images of a tight
reservoir sample. The distribution of these domains could be extrapolated by
calibration against size distributions measured in FIB-SEM images. For this,
we assumed a mean permeability for the dominant clay mineral (illite) in the
rock and assigned it to the microporous domains within the structure. The
results prove the applicability of our novel approach by combining FIB-SEM
with X-ray tomographic rock core scans to achieve a good correspondence
between measured and simulated permeabilities. This methodology results in a
more accurate representation of reservoir rock permeability in comparison to
that estimated purely based on µXCT images.
Introduction
Depositional environment and subsequent diagenetic alterations are two key
factors that influence the bulk mineralogical composition and the authigenic
clay mineral inventory of a reservoir
(Wilson and Pittman, 1977; Worden and
Morad, 1999), and therefore the fluid-flow properties of the porous rock. A
well-established technique to image and analyse rapidly the 3D physical
properties of porous rocks is computer X-ray microtomography (µXCT)
combined with the concept of digital rock physics
(Andrä et al., 2013a, b;
Okabe and Blunt, 2004). By applying monochromatic synchrotron radiation, it
is possible to overcome conventional µXCT artefacts like beam
hardening and problems that arise due to limited phase contrast, lack of
resolution and edge preservation, as well as low signal-to-noise ratios
(Brunke
et al., 2008; Kling et al., 2018; Lindquist et al., 2000; Mayo et al., 2015;
Spanne et al., 1994). Synchrotron-based µXCT scans with image
resolutions in the order of 1 µm can provide a sound basis for flow
and transport modelling of tight sandstones as suggested by
Peng et al. (2014). They found that synchrotron µXCT imaging is necessary for tight sandstones when the connectivity of the
pore space is low and pore throats cannot be resolved using a conventional
µXCT scanner. They further concluded that a high abundance of the
smallest resolvable pores falsifies modelled permeabilities due to an
overestimation of actual pore sizes. Several studies have shown
sub-micrometre pore structures to be a frequent feature of tight reservoir
rocks (Jiang, 2012; Shah et al., 2016; Soulaine et al., 2016). Most of these nanostructures are related to
different types of clay minerals; most commonly illite, kaolinite, chlorite
and smectite (e.g. Wilson and Pittman, 1977;
Worden and Morad, 1999; Desbois
et al., 2016).
Although known for years, considering such structural features below µXCT resolution in pore-scale models remains challenging
(Alyafei
et al., 2015; Guan et al., 2019; Liu and Mostaghimi, 2018; Liu et al., 2020;
Menke et al., 2019; Peng et al., 2012). Soulaine et al. (2016)
systematically analysed the effect of sub-resolution domains with varying
permeabilities on the simulated permeabilities of Berea sandstone (20 vol % porosity, 2 vol % sub-resolution domains) and found that
calculated permeabilities can be reduced by up to 50 %, if microporous
domain permeabilities converge towards zero. Thus, it is evident that
neglecting sub-resolution information can lead to a significant
overestimation of rock permeability in such simulations
(e.g. Saxena et al., 2018, 2017). Menke et al. (2019) utilized a multi-scale Brinkman area approach, applying different
permeabilities for each microporous domain to simulate flow in
mono-mineralic carbonate rock. They showed that Stokes–Brinkman models are
in good agreement with experimental data, whereas Stokes and/or Navier–Stokes
models alone were not able to predict permeability in a conventional flow
scenario. They also demonstrated that for pure carbonates, a direct
correlation can be established between observed density contrasts and
specific physical properties, such as porosity and permeability. However,
this approach is not applicable in a system with more than one rock-forming
mineral, such as a tight sandstone, where density contrasts relate to
different mineral phases as well as sub-resolution porosities.
The lack of distinct material information for a voxel is often ascribed to
the partial volume effect (e.g. Kessler et al., 1984; Ketcham and
Carlson, 2001). To overcome the impact of this effect on physical properties
of a rock, imaging techniques that can resolve the pore structure at
different length scales have to be applied. For estimating the permeability
of reservoir rocks, the resolution achieved by synchrotron radiation imaging
lies within an acceptable range (Saxena et al., 2018).
Several studies have demonstrated that, by combining X-ray and scanning
electron imaging, the pore space of tight clay-bearing rocks can be
spatially resolved from the millimetre down to the nanometre scale (e.g.
Desbois et al., 2016; Hemes
et al., 2015; Markussen et al., 2019).
In this study, we aim to demonstrate a new approach by combining
synchrotron-based µXCT imaging with focussed ion beam scanning
electron microscopy (FIB-SEM) to improve flow simulations in a tight
sandstone formation with high clay mineral content. First, we used machine
learning-based image segmentation to enhance pore space segmentations of
artefact-rich FIB-SEM topologies. Then, we conducted Navier–Stokes
simulations on FIB-SEM topologies. Finally, we subsequently used these
simulations as input data for sub-resolution domains in µXCT-based
Stokes–Brinkman models (Brinkman, 1949; Neale
and Nader, 1974). This novel morphology-based approach for
sub-resolution-rich materials results in simulated permeabilities that fit
experimental results significantly better than using Navier–Stokes
simulations alone.
Sample preparation and characterization
In this study, a well-characterized illite-bearing Upper Rotliegend
sandstone from Germany was used, which was sampled from the quarry
Schwentesius near Bebertal village (Heidsiek et al., 2020).
This well-known location exposes an analogue of the Permian gas reservoir
sandstone of the Flechtingen High, which formed part of the North German
Basin. Thin section analyses of this reservoir rock have shown that illite
is the main clay mineral, which primarily occurs as a coating along detrital
grains, illite meshworks grown on coatings, diagenetically altered
K-feldspar and illitized lithoclasts
(Fischer et al., 2012).
Samples were taken from a large sandstone block displaying a variety of
different aeolian and fluvial depositional facies (Fig. 1a). We identified
several facies denoted A to J from four different types of sedimentary
depositional layers. The samples of the different facies were numbered from
left to right. Samples were drilled out and extracted from marked locations
in the form of plugs with a diameter of 2.5 cm. The plugs were cut into
three segments which were used for X-ray diffraction analyses, helium
porosimetry measurements and FIB-SEM imaging. Mini plugs were drilled
directly beside the plugs with a diameter of 3 mm and a length between 10
and 20 mm. The larger plugs were used for mineralogical and geochemical
analysis, and to measure permeabilities experimentally. The mini plugs were
scanned at the synchrotron radiation-based µXCT imaging beamline P05
at the PETRA of DESY Hamburg (Germany). After the synchrotron measuring
campaign, the mini plugs were additionally examined by FIB-SEM and energy
dispersive X-ray spectroscopy (EDX) imaging to obtain qualitative and
quantitative information about the clay mineral particles found within the
rock pore space. Microporous structures in the Rotliegend sandstone sample
could be resolved by comparing µXCT and FIB-SEM images. The term
“microporous” refers to the definition of sub-micrometre porosity by
Soulaine et al. (2016), who differentiated between void,
solid and microporous voxels in µXCT images. According to for
example Tinet et al. (2020), the term “nanoporous” is used to describe
structures with predominant pore sizes in the nanometre range (0.2–1000 nm). In this range, the clay minerals often appear as a greyish pore-filling
phase in µXCT images.
(a) Upper Rotliegend sandstone block showing four main deposition
facies. (b) Sampling locations and sizes of the extracted plugs (green) and
the mini plugs (red) extracted.
X-ray diffraction analyses using the Rietveld analysis program Profex 4.0
(Doebelin and Kleeberg, 2015; Ufer et al., 2012) have shown a homogenous mineralogical composition along the layers of
the sample block with only slight variations in content (Fig. 2).
XRD analysis of the main minerals in the plugs based on Rietveld
calculations, determined for several samples of facies A to J of the
sandstone block.
The main components are quartz (58 wt %–69 wt %), authigenic and diagenetic
feldspars (12 wt %–20 wt %), calcite (1 wt %–18 wt %), and illite (10 wt %–17 wt %). In the <0.2µm size fraction, we observed traces
of swelling smectite interpreted as contaminants from surface weathering.
Accessory hematite and barite particles also occur with abundances of
<1 wt %. In the Bebertal tight sandstone, illite is by far the
most abundant clay mineral and makes up ∼ 95 wt % of the
<2µm fraction. Commonly, the nanometre-scale microstructural
features of reservoir rocks containing significant amounts of clay minerals
are usually not detectable at µXCT resolution due to a low absorption
contrast (Ahmad et al., 2018).
Permeability measurements and pore size distribution
A helium gas-driven permeameter under steady-state conditions was used to
experimentally obtain permeability values for the individual plugs (e.g.
Filomena et al., 2014). Compressed air
was used to apply a pressure of 10 bar to the core samples that were coated
by a latex membrane. A Bronkhorst F-111C-RBB-33-V was used as a flowmeter.
It was calibrated by polynomial calibration and has a measuring accuracy of
±0.5 % plus ±0.1 % of the full scale. To measure the
differential pressure, an Emerson Rosemount 3051S Ultra was used with an
accuracy of ±0.045 %. To control the inflow and outflow, two
pressure sensors (Brooks 5866) with an accuracy of ±0.5 % were
installed. For permeability measurements, the inflow and outflow pressures
of the helium flux were sequentially increased in up to six pressure steps.
The differential pressure was kept constant at 200–500 mbar depending on
the sample properties. The intrinsic sample permeability was derived from
the apparent gas permeability, Kg, determined for each pressure step
using Darcy's law (Liu et al., 2017):
Kg=2Qp2ηLA(p12-p22),
where Q is the measured gas flow rate, η is the dynamic viscosity
of the permeant, L is the sample length, A is the sample cross section, and
p1 and p2 are the inflow and outflow pressures. By plotting
Kg against 1(p1+p2)/2, the data can be fitted by a
straight line. The intercept of the best-fit line at the Kg axis
corresponds to the intrinsic sample permeability, Kint
(Gao and Li, 2016; Klinkenberg, 1941). Also, MICP
measurements were conducted with an Autopore IV Series (Micromeritic
Instrument Corp.) to determine the pore size distribution of a dried
sub-sample with a weight of ∼ 2.5 g, which was taken from a
cross-bedded aeolian layer of the sandstone block. Based on the capillary
law, MICP enables the analysis of a wide spectrum of pore sizes (3 nm to
>900µm), corresponding to a pressure range of 0–414 MPa. As a non-wetting liquid with a high contact angle (130–140∘), mercury only penetrates a pore when pressure is applied. Under the
assumption of cylindrical pores, the applied pressure is directly
proportional to the pore throat diameter as described by the Washburn
equation (Washburn, 1921):
D=-4γ⋅cosθP,
where D is pore throat diameter, γ is the surface tension, θ is the contact angle and P is the applied pressure.
Synchrotron-based µXCT
For synchrotron tomography, a beam energy of 29.87 KeV was used while the
sample-detector distance was 1.2 cm. The effective image resolution of the
detector equipped with a CCD camera was 1.22 µm per pixel, while the
image size was 3056×3056 pixels. We used an advanced
reconstruction script described in Moosmann et al. (2014) with the
MATLAB® software and binned the images by a
factor of 2 before reconstruction to increase the signal-to-noise ratio.
This decreased the voxel resolution to 2.43 µm and changed the image
size to 1528 × 1528 pixels. The number of projections was 1200,
with the information of five subsequent images used to calculate an average
for every projection image. After reconstruction of the 3D image stacks, the
scans were denoised using the non-local means filter of the GeoDict 2020 software
package (Buades et al., 2011). Image segmentation of the
mini plugs was realized by conventional greyscale thresholding. The
greyscale values for each phase on a 16 bit intensity range were 0–8550
for pores, 8551–12 880 for grains, 12 881–25 000 for cements and 25 001–65 535 for high-density cements and oxides. Since the best threshold values
for each phase varied slightly from sample to sample, these values were
adjusted for each segmentation within a small range of ±50. A
qualitative comparison with machine learning segmentation revealed a better
pore-to-solid segmentation and resolving of small pore throats by
thresholding (Fig. 3). Since the main goal was to achieve the best possible
permeability estimation, the differentiation between pore and solid is more
important for permeability estimation than the accurate segmentation into
different phases (Khan et al., 2016).
Leu et al. (2014) point out that even a small variation
in pore throat morphology can have a large impact on the estimation of
permeability.
(a) Greyscale 2D µXCT image of a mini plug. (b) Segmented
phases using multi thresholding. (c) Segmented phases using the machine
learning image classification module of the software ilastik (Version 1.3.3) by Berg et al. (2019).
For permeability simulations, we either used the fast Fourier transformation
(SIMPLE-FFT) or the left-identity-right (LIR) solvers, both implemented in
the FlowDict module of the GeoDict software package
(Linden et al., 2015; Moulinec and Suquet,
1995). While the SIMPLE-FFT solver is fast for calculating low-porosity
domains, the LIR solver is better suited for high-porosity domains and
requires less memory. Both iterative finite volume solvers can apply
Navier–Stokes and Navier–Stokes–Brinkman equations. Derived from these
equations, Darcy's law (Eq. 1) is used to calculate the permeability of a
material (Darcy, 1856):
u=-Kη(∇p-f).
In Eq. (3), u is the three-dimensional average fluid-flow velocity, K is
the permeability, η is the fluid viscosity, p is the intrinsic average
pressure tensor and f is the force density field, which was defined using
the Navier–Stokes conservation of momentum equation for all three
dimensions (Eq. 4):
-ηΔu+u⋅∇u+∇p=f.
The Brinkman term can be added to the Navier–Stokes equation where porous
voxels are required. These voxels include the nanoporous flow resistivity:
-ηΔu+u⋅∇u+ηK-1u+∇p=f,
where K-1 is the inverse of the permeability tensor and ηK-1
the flow resistivity. The applicability and robustness of combining
Navier–Stokes equations with the Brinkman term has been validated by
Iliev and Laptev (2004).
We calculated the permeability with symmetric boundary conditions in
tangential and flow direction with a pressure drop of 20 370 Pa. The
symmetric boundary conditions are valid for low-porosity structures with
non-periodic pore throat geometries. The differential pressure value was set
to be able to compare the results with helium permeation flux measurements
where similar values have been used. As a convergence stopping criterion, a
low error bound of 0.05 was chosen. The error bound criterion uses the
result of a previous iteration to predict the final solution by linear and
quadratic extrapolation. This stops the iteration when the relative
difference regarding the prediction is smaller than 5 %. During this
process, the solver recognizes oscillations and local minima or maxima which
prevents a premature stopping of the solver during the iteration process.
FIB-SEM measurements
In this study, a Zeiss Auriga crossbeam electron microscope equipped with a
Gemini electron column and an Orsay Physics ion beam was used. SEM images
were taken at 1 kV with an in-lens secondary electron (SE2) detector, and
FIB slicing was executed with a beam current between 0.5 and 2 nA and a
voltage of 30 kV. This resulted in a slice thickness of 25 nm. A large FOV
of ∼ 20 µm could be reached. To derive structural
information from the FIB-SEM images, extensive post-processing of the data
was required. Following image alignment and cropping, stripes and shadow
artefacts were filtered out before image segmentation. The Slice Alignment operation of the
module ImportGeo of the GeoDict 2020 software package was used to align the images, while the
Curtaining Filter was used for stripes correction (Fig. 4). In general, the segmentation of
pores in FIB-SEM images was not straightforward since scans of porous
polished sections are pseudo-2D and contain information from behind the
actual imaging plane (De Boever et al., 2015).
(a) Raw SEM image of secondary illite growth in a porous feldspar
with streaks and shadow artefacts. (b) Images with aligned slices of a
cropped region of interest. (c) Filtered greyscale SEM image of the illite
meshworks. (d) The binarized result after image segmentation with pixel
classification algorithms of the software ilastik.
As multi-thresholding and watershed segmentation algorithms have problems
with shine-through artefacts (Prill et al., 2013),
capturing the correct 3D pore space geometry is of crucial importance for
the determination of a realistic permeability. Recent advances have shown
that machine learning image segmentation software can successfully be
utilized to segment pore space in CT scans
(Berg et al., 2018).
The software ilastik, an interactive learning and segmentation toolkit by
Berg et al. (2019), was used for the interactively
supervised machine learning segmentation of the phases in our FIB-SEM images
(Fig. 4). The built-in pixel classification module groups the probability of
each pixel to be assigned to a phase according to their different imaged
features. The software provides a set of different images based on features
derived from the original image (filtering and smoothing, edge detection,
etc.). Examples of the correct class (phase) provided by the user builds a
decision surface in feature space. Based on these features, a random forest
classifier assigns each pixel or voxel to a certain phase. This method works
for both 2D and 3D data. The accuracy of this method increases with the
number of user-provided training data. In a manually controlled workflow, it
was possible to reach high segmentation accuracies with only minor over- or
underestimations of the pore space (Fig. 4c, d).
Defining microporous domains
The need to define microporous domains results from the mismatch of
permeability between µXCT simulations and gas-driven permeameter tests.
While the simulation of permeability in structures with high permeability
and porosity obtained by µXCT scans is precise, the effect of
nanoporosity below resolution on permeability increases with decreasing
permeability (Pittman and Thomas, 1979; Saxena
et al., 2018). When comparing backscattered electron (BSE) images with µXCT images of the same slice, it becomes apparent that the smallest pores
in µXCT images simplify the real pore structure (Fig. 5).
Furthermore, SEM and EDX images revealed that most of the pores are filled
with clay minerals. Since both void and microporous regions share similar
greyscale values, it is impossible to correctly differentiate upon
segmentation. In this approach, the segmentation of the pores includes clay
minerals with a low absorption contrast. To determine the distribution of
illite in µXCT scans, we used correlative µXCT, SEM and EDX
measurements. For this, a 3 mm plug was embedded into epoxy resin and then
ground and polished until the region of interest was reached. Two sites were
chosen for EDX mappings (Fig. 5). Comparing EDX mappings with CT images
shows that the distribution of illite agrees with the textural findings of
Fischer et al. (2012).
(a)µXCT and corresponding (b) BSE image of the polished
section including the close-up regions S1 and S2 that were used for EDX
mapping to analyse the distribution of illites. (c) Ternary plot of Ca, K
and Al content of each pixel in the mappings. Calcite, illite and K-feldspar
plot in three distinct regions. By selecting these areas, a phase map overlay
was created that visualizes the location of these phases in SE images. (d)
Region S1 and S2 with phase map overlays. Illite is predominantly found in
the vicinity of altered feldspars, as coating along grain boundaries and as
authigenic pore filling.
Furthermore, it becomes apparent that illite enrichments coincide with
regions that are usually referred to as void pore space in µXCT
images. The mismatch between real pore structure and segmented pore space is
highest in small pores and throats. An average size of ∼ 5 µm was measured for illites which cover the surfaces of the quartz
and feldspar grains. This corresponds to approximately two voxels in the
µXCT images. To refine flow and reduce the influence of overestimated
pore sizes in these specific regions, we define all pores with a diameter
≤2 voxels as microporous domains. The Brinkman term accounts for small
pores, where grains are porous themselves (Brinkman, 1949).
To extract these regions from the initial pore space segmentation Fp∗ we calculated Euclidian distance maps as used by
Maurer et al. (2003):
d=x2-x1+y2-y1+(z2-z1),
where d is the distance between two points; x1, y1
and z1 are the coordinates of the first point; and x2,
y2 and z2 are the coordinates of the second point. By
selecting d(≤2) all pore voxels with the nearest distance of ≤2
voxels to the next solid surface are masked, including the outer rim of
larger pores (Fig. 6b). To obtain a mask with pores ≤2 voxels, this rim
layer surplus must be removed. This was realized by performing a
morphological dilation operation on d(>2) using a structure element SE2 of
23 voxels. By subtracting the dilated image from d>2, the microporous domain data Fμ2 (Fig. 6c) can be
obtained.
Fμ2=d≤2-δSE2d(>2)
(a) Segmented µXCT image of the pore space of a mini plug.
(b) Image with applied microporous domain using the Euclidian distance map
approach. (c) Image with removed outer rim layer of larger pores.
In GeoDict the permeabilities of the microporous domains are calculated with the
Brinkman term. Based on flow simulations on 3D FIB-SEM images of illite
meshworks, an isotropic permeability was assigned to the microporous domains
(Fig. 6). Yoon and Dewers (2013) confirmed the validity of
the approach of extrapolating structural features of clays measured by
FIB-SEM to the pore scale. The complete workflow for a precise permeability
simulation is illustrated in Fig. 7.
Workflow for microporous domain modelling combining FIB-SEM
nanoporosity with µXCT scans. Sample E9 of the aeolian layer with
well-distributed flow paths was selected for visualization. Upon the
modelling of the microporous domains, the total fluid-flow velocity
decreases. Displayed simulated permeabilities include the mean values of
five samples with the best distribution of percolation paths. Fluid-flow
velocities close to zero are transparent.
Results and discussionPermeability measurements and pore size distribution
Four to six pressure stages were applied to the plug samples to determine
their gas permeability. In Table 1, parameters which were used for the
permeameter test are shown for sample F8. These values yield a
Klinkenberg factor of 0.56 and a quadratic regression coefficient of 0.94.
Parameters of the gas permeameter flux experiment (sample F8),
including measured helium gas flow rate, recorded inflow and outflow
pressures as well as calculated apparent gas permeability for each pressure
stage. For evaluating the intrinsic sample permeability, the sample
cross section (0.025 m) and length (0.05 m) as well as the dynamic viscosity
of helium (18.4 × 10-6 Pas) is also needed.
Overall, the measured intrinsic permeabilities range between 1.1 and 5.4 mD with an average value of 2.9 mD for both aeolian facies (Table 2; Fig. 1,
sample series E and F). The quadratic regression coefficient
ranges from 0.732 to 0.998, which indicates a high accuracy of the
permeameter test. The small variation of the physical rock properties
between the main layers is induced by different angles of the grain layering
in each plug sample and small-scale variations in grain and pore size
distributions. The observed variance is in the typical range of observed
permeability fluctuations for tight reservoir sandstones
(Lis-Śledziona, 2019). Considering the observed
permeability, the studied rock samples are at the lower end of the
permeability range known for sedimentary reservoir rocks (Gluyas
and Swarbrick, 2004).
Intrinsic permeabilities and porosities of the measured sandstone
plug samples.
The pore size distribution of the sub-sample analysed by mercury intrusion
porosimetry (Sect. 3.2) was obtained by a semi-logarithmic representation
of the normalized intrusion volume achieved per pressure interval (Fig. 8).
The consumed capillary stem intrusion volume of 27 % was within the
acceptable range needed for precise measurements. We discovered a major peak
at approximately 1 µm, representing the most common pore throat
diameter.
Distribution of the pore throat diameters of a sample of the
aeolian facies in layer J obtained from mercury intrusion porosimetry
(MICP).
In addition to the mercury porosimetry results, calculations of the pore
size distribution of FIB-SEM and µXCT mini plug images were conducted
using the GeoDict module PoroDict. This module provides an algorithm that virtually pushes
spheres of different sizes into a medium to determine the 3D pore size
distribution (Münch and Holzer, 2008). FIB-SEM
measurements of the pore size distribution cover the small pores related to
the illite nanoporosity, while µXCT porosimetry illustrates the pore
size distribution of the larger-scale intergranular pore space skeleton.
The µXCT pore volume analysis shows a constant rise towards the
smallest observed pore diameters (2.44 µm; Fig. 9). The steepest
slope is observed between 7.3 and 12.2 µm, which indicates that the
largest volume of pore bodies is observed in that range. The calculated pore
size distribution from nanoporous 3D structures of illite meshworks obtained
by FIB-SEM imaging shows three distinct peaks at 75, 125 and
250 nm. This illustrates that the most common inscribed pore diameters are
relatively small compared to the actual extent of the pores, which is
considered typical for slit-shaped pore systems where the pore axial ratio
is high (Desbois et al., 2016). Ultimately, this results
from the different thicknesses of clay platelets and heterogeneity in the
alignment of the illite nanostructure (Aplin et al., 2006).
Pore throat diameters from µXCT (a) and FIB-SEM images
(b).
An apparent gap exists between diameters observed by MICP and 3D imaging
with FIB-SEM and µXCT. While MICP peaks at around 1 µm,
diameters observed by µXCT start at 2.4 µm and the largest
inscribed diameters observed by FIB-SEM are below 700 nm. However, it must
be noted that pressure-controlled MICP generally gives information about the
dimension of pore throats, whereas pore size distributions provided by
imaging techniques give information about pore body volumes
(Zhao et al., 2015). Furthermore, pore shielding may cause
an underestimation of larger pores for MICP
(Abell et al., 1999; Gane et al., 2004).
The occurrence of authigenic illites is the likely cause of this effect as
they are commonly found in pore throat areas. Since the Washburn equation
assumes ideal pore throats of cylindrical shape, the underestimation of
larger pores becomes more evident with the increasing complexity of the pore
throat system at both the millimetre and nanometre scale (Washburn, 1921).
Therefore, a direct comparison between the used methods is unlikely to
result in compatible results.
Permeability simulation
A calculated isotropic permeability of 2.53×10-3 mD
was used for the microporous domains based on the Navier–Stokes fluid-flow
simulations of permeability of FIB-SEM scans of the illite meshworks. The
number of porous voxels resulting from clay mineral modelling ranges between
3.3 vol % and 7.1 vol % of the total structure volume for all µXCT scans considered. A comparison of the modelled clay minerals in
µXCT scans with XRD mass balancing highlights a large difference
between the mineral abundances measured (Table 3, Fig. 2).
Modelled clay mineral content within the microporous domains in
µXCT scans.
C3D2D9E9F7G2G8H9I3I9J3J7MeanStandarddeviation (±1σ)Clay mineral content (vol %)4.73.35.05.47.15.06.06.34.16.26.26.15.41.0
While the mean amount of clay minerals based on XRD measurements was 12.7 wt % (about 11.3 vol % within a structure with 8 % porosity), an
average amount of 5.4 vol % was modelled by the distance map algorithm.
This is expected since the illite content inside grains was not modelled
since it has no effect on permeability. We simulated permeability of 12 mini
plug samples that were scanned by µXCT and compared them with
measurements from gas permeameter experiments. As a first step, we extracted
and illustrated the 10 largest and 100 largest open flow paths through the pore
space of all mini plug cores before the modelling of the microporous domains
(Fig. 10). This yields information concerning the heterogeneity of the flow
fields and allows the validity of the Navier–Stokes simulations to be checked.
Structures with flow impingement often cause numerical problems, which
results in an artificial underestimation of the permeability simulations.
Significant underestimations of permeability after clay mineral modelling
were also found in areas where the percolation paths in the samples were
limited to a few voxels in the structure before modelling.
Comparison of percolation paths in two µXCT reconstructions
of the 3D imaging data without microporous domain modelling. (a) Well
distributed percolation paths in a µXCT reconstruction of mini plug
sample E9 without microporous domain modelling. (b) Constricted percolation
paths with flow impingement limited to a small region of the structure in
mini plug sample I3 without microporous domain modelling. The 10 largest
percolation paths are coloured in purple; the 100 largest percolation
paths are coloured in cyan.
This effect leads to an artificial permeability drop, which renders
calculations less precise. Since evenly distributed flow paths are
necessary to determine the true permeability of a volume of a rock, we
considered only permeability calculations of samples which show no flow
impingement for modelling (Bear, 1972;
Leu et al., 2014; Zhang et al., 2000). The validity of the simulations was
ensured by applying the variance algorithm on the flow fields discussed in
Jacob et al. (2019).
The mean value of the experimentally obtained intrinsic gas permeabilities
was 2.9 mD (Table 2, Fig. 11a). Samples with no flow impingement had an
initial mean permeability of 26.5 mD, whereas the mean permeability
simulated on µXCT images was 26.5 mD (Fig. 11b). With applied
microporous domains, a mean permeability of 1.9 mD was calculated (Fig. 11c).
The mismatch between measured and simulated permeabilities could be
decreased to 1 mD (-34.5 %) compared to 23.6 mD (+813.8 %) prior to
the modelling. Hence, our approach significantly improved the match between
measured and simulated permeabilities and lies within the standard deviation
of the permeabilities measured by the gas permeameter (Table 2). Histograms
of the simulated fluid-flow velocities indicate a strong decrease in
velocities due to the modelling of the microporous domains in sample E9
(Fig. 12). Correlation histograms of the velocities depict differences
induced by the modelling. While identical fluid-flow field velocities would
plot as a single straight line, the velocities, in this case, show straight
lines with varying slopes. This indicates a general decrease in the
fluid-flow velocities with a splitting of different flow path velocities.
Points which plot as a line represent a main fluid-flow path with a direct
correlation of the velocities between the original and the modelled
structure's flow field. Furthermore, a wider spread of the distribution of
high velocities results from microporous domain modelling. This indicates
narrow pore throats where fluid-flow velocities are locally enhanced
compared to the structure before the modelling.
Comparison of measured and simulated permeabilities. The boxes
indicate the upper and lower quartile of derived permeability values. Median
values are coloured in red; mean values are coloured in black. (a) Measured
permeabilities using the gas permeameter test. (b) Simulated permeabilities
using µXCT data. (c) Simulated permeabilities using µXCT data
with modelled microporous domains.
Comparison of unmodelled µXCT images and modelled
microporous domain images (a). Correlation histogram prior to and after the
modelling that shows a decrease in the main fluid-flow velocities, while few velocities after the modelling were increased with a wide spread of
distribution (b).
Based on our combined analytical and numerical study, further research may
help to increase the accuracy of simulated permeabilities even further.
Since isotropic permeabilities of the microporous domains were applied to
µXCT images, the accuracy can be improved by taking the anisotropy of
clay mineral fabrics and surface topology into account. This can be done by
applying anisotropic permeabilities in the calculation of the microporous
domains. While this study showed a good match between the experimental and
simulated permeability, the need to include heterogeneities of clay mineral
layering to improve the simulations was depicted in
Villiéras et al. (1997).
The presented approach might also be applicable to other rock types such as
mudrocks, carbonates, etc., if the preconditions are met. The examined rock
properties (e.g. permeability) should be based on the same scale
dependencies as the rock analysed in this work (e.g. Grathoff et al., 2016).
Conclusions
Overall, the outcome of this study shows that combining µXCT and
FIB-SEM imaging with numerical models constitutes a valuable and novel
approach for determining physical properties of clay-bearing tight reservoir
rocks. Considering the high number of accessible pores in the scans, the
phenomenon of flow impingement was mainly attributed to the unresolvable
nanoporosity. While permeability, which is one of the most important
reservoir properties, is often determined by simulations based on µXCT scans of small samples taken from a field-scale reservoir, we were able to demonstrate that an accurate estimation for clay-rich and low-permeability
rocks is only possible if nanoscale porosity is also included. Thus, our
simulations using the Euclidian distance map approach resulted in an
improved match with stationary gas permeameter measurements in contrast to
permeability simulations merely based on unmodified µXCT images.
Adopting this multi-method approach, we increased the accuracy of simulated
permeabilities of samples measured by µXCT. These results have
important implications for improved modelling of reservoirs relevant to gas
and water applications. A realistic simulated permeability of a tight
reservoir sandstone could only be achieved by appropriate modelling of the
nanoporosity related to matrix clay minerals (illite) that occur below the
µXCT resolution. The simulated permeability based on combined µXCT and FIB-SEM images and modelled microporous domains showed good
agreement with the experimental results. Obtaining an even distribution of
the simulated fluid-flow paths through the sample without flow impingement
was necessary to obtain an accurate permeability estimation from 3D imaging.
Resolving the nanopore structure and distribution of clay mineral-related
features by the combined analytical and numerical modelling approach
represents a tool for achieving a more accurate understanding of the fluid-flow behaviour within tight sandstones, with direct relevance to predicting
the injection, storage or extraction of gas or water in a reservoir rock.
Our multi-method approach can be applied to determine more accurate
permeability values and flow paths for reservoir rocks with high clay
mineral content if direct experimental measurements are not successful.
Hence, future studies should focus on distinguishing the different
morphologies of clay minerals and their related anisotropic effect on rock
permeability. The permeability of the nanoporous structures depends
highly on the layering of the clays and their spatial orientation on the
grain surfaces and within feldspars. This approach should include a variety
of various sedimentological facies, also with high porosity and permeability,
to investigate whether clay mineral modelling is also a valid tool for such
sedimentary rocks.
Data availability
Data can be partially accessed upon request by Arne Jacob, Markus Peltz and Frieder Enzmann.
The supplement related to this article is available online at: https://doi.org/10.5194/se-12-1-2021-supplement.
Author contributions
The conceptualization was developed by AJ, MP, SH, FE, LNW, GG, PB and MK. The process methodology was developed by AJ and MP. The software was provided by FE while the programming was conducted by AJ and MP.
The final validation of the results was done by AJ, MP, SH, FE, LNW, GG, PB and MK. Formal analysis was executed by AJ and MP. The review and editing was executed by AJ, MP, SH, FE, OM, LNW, GG, PB and MK. The visualization of the results was conducted by AJ and MP. Supervision was provided by FE, LNW, GG, PB and MK. The project was administrated by FE, LNW, GG, PB and MK.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
This work was supported by the German Federal Ministry of Education and
Research (BMBF) “Geological Research for Sustainability (GEO:N)” program,
which is part of the BMBF “Research for Sustainable Development (FONA3)”
framework program. It is part of the project ResKin (Reaction kinetics in
reservoir rocks, 03G0871E). We would like to thank Fabian Wilde and the
staff of PETRA synchrotron facility at DESY Hamburg for their assistance with
the imaging beamline P05. Jens Hornung and Meike Hintze from the TU
Darmstadt are also acknowledged for providing us with gas permeability
measurements of the studied samples.
Financial support
This research has been supported by the German Federal Ministry of Education and Research (BMBF) (grant no. 03G0871E).This open-access publication was funded by Johannes Gutenberg University Mainz.
Review statement
This paper was edited by Florian Fusseis and reviewed by two anonymous referees.
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