Modern estimation of rock properties combines imaging with advanced numerical
simulations, an approach known as digital rock physics (DRP). In this paper
we suggest a specific segmentation procedure of X-ray micro-computed
tomography data with two different resolutions in the
Three-dimensional information on rock microstructures is important for a
better understanding of physical phenomena and for rock characterization on
the microscale. Various destructive and non-destructive methods for
obtaining a 3-D image of the rock microstructure exist (Arns et al., 2002;
Saenger et al., 2004; Madonna et al., 2013; Cnudde and Boone, 2013; Wang et al., 2013, and
references therein). The most common non-destructive 3-D imaging technique for
rock samples is X-ray computed tomography (XRCT). A common problem, however,
is a clear trade-off between sample size and resolution. For each material a
specific and large enough sample size is required to ensure that the
selected volume is representative of the physical property to be computed
(e.g. Hill, 1963; Costanza-Robinson et al., 2011; Andrä et al., 2013a).
It can, however, be at the expense of a loss of pore feature resolution. In
the last decade, the X-ray micro-computed tomography (micro-XRCT) method
became widely available and many modern studies have made use of it to obtain
3-D rock images (e.g. Fusseis et al., 2014). The resolution of micro-XRCT of
up to (0.6
Rock images from 3-D XRCT can be used for predicting properties such as
porosity, permeability, pore size distribution, effective elastic moduli, or
electrical conductivity (e.g. Andrä et al., 2013b). For example, Darcy
permeability can be predicted by numerically simulating single-phase fluid
flow through 3-D rock pore structure models, with the numerical results being
in reasonable agreement with laboratory measurements (e.g. Osorno et al.,
2015; Saenger et al., 2016). In this case, the resolution of the micro-XRCT
technique is sufficient, because fluid pathways predominantly follow larger
pores. However, if the pore size is much smaller than 1
An example of such a disagreement between laboratory and digital rock physics (DRP) estimates is described in Andrä et al. (2013a, b). In these benchmark papers a comparison between different numerical methods is presented. All DRP estimates of the effective elastic bulk modulus use the same segmented data set. Regardless of the numerical approach, all computational predictions overestimate the bulk modulus measured in the laboratory. This conclusion is mostly based on Berea sandstone although carbonates are considered in this study. However, Jouini et al. (2015) also reports about an overestimation of effective elastic properties of carbonates by DRP. Therefore we conclude here that the digital rock images themselves and/or the computational workflow have to be improved to provide better estimates of effective properties of rocks. In this paper we consider a carbonate data set in detail and suggest techniques to achieve a better agreement between numerical predictions and laboratory measurements. Our study is complementary to the DRP carbonate studies performed in Derzhi et al. (2010), Lopez et al. (2012), Ringstad et al. (2013), Andrä et al. (2013a, b), and Jouini et al. (2015). In contrast to these studies our digital rock physics study is complemented with a very detailed experimental characterization (Sect. 2). Our suggested segmentation technique (Sect. 3) is used to estimate effective mechanical and transport properties (Sect. 4). Among others, we observe a two-phase trend for monomineralic (calcite) carbonates which can be regarded as an upper bound for velocities at all scales (see discussion in Sect. 5) due to the observed self-similarity of those rocks (Jouini et al., 2015).
We studied samples of two carbonates from the Upper Cretaceous carbonate system of the Gargano–Murge region (southern Italy). Carb-A is a limestone from the Paleocene–Eocene Peschici Formation, and Carb-B is a micritic mudstone from the Late Cretaceous Monte Acuto Formation (Martinis and Pavan, 1967; Cremonini et al., 1971). Both carbonates are composed of nearly 100 % calcite (Scotellaro et al., 2014).
Both samples have been characterized in the laboratory in detail in Vialle
et al. (2013), e.g. SEM images showing the microstructure of these two
samples. We only give a short summary here. Both carbonate samples display a
matrix of micrite (abbreviation for microcrystalline calcite) with a typical grain
size of 1–4
He-grain density, bulk density, and resulting porosity, as well as
air permeability, were previously measured at room pressure and temperature
on core plugs 0.025 m in length and diameter (Vialle et al., 2013).
The associated errors did not exceed 0.5, 1, and 2 %,
respectively.
Helium bulk and grain density (in g cm
Pressure dependence of the elastic wave velocities for the two
studied samples. Pressure is in megapascal, and
Nanoindentation tests were performed to obtain stiffness (Young's modulus)
of the carbonates at the micrometre scale. These tests were performed on
room-dried samples consisting of two small irregular pieces, about 5 mm
thick and with a surface of a few cm
Typically, the extraction of the mechanical properties is achieved by using
the
Data were further corrected considering deviation of the indenter tip from ideal geometry, initial penetration into the rock below a load threshold, and compliance of the loading column, leading to a nominal uncertainty of indentation moduli of < 2 GPa.
Young's moduli, E, can be calculated from the indentation moduli
(Fischer-Cripps, 2004) according to
Indenter properties are
Nanoindentation results for Carb-A (left-hand side) and Carb-B (right-hand side). In blue we illustrate the corresponding moduli range from ultrasonic experiments on dry samples from 0 to 30 MPa confining pressure, and in green we illustrate the moduli range given by the solid anisotropic calcite crystal. Overall we observe that the medium effective indentation module M is slightly stiffer for Carb-A (26 GPa vs. 25 GPa).
Two samples were prepared for imaging with micro-XRCT from the cuttings, one
from Carb-A and one from Carb-B. A cylindrical-shaped sample of 1.5 cm in
height and 2 mm in diameter was achieved by gently grinding the cuttings,
first on the side on a rock saw blade, then by hand using sand paper
(grit 120). This procedure allows us to obtain very thin cylinders while
minimizing mechanical damage that classical drilling would produce. These
cylindrical samples were then glued with Crystalbond509 (SPI suppliers) on a
2 mm diameter flat-head metal pin, which was itself inserted in the core
holder of the micro-tomograph. The 3-D X-ray Microscope Versa XRM 500 (Zeiss
– XRadia) was used with a X-ray energy of 60 keV. Two different settings of
source-to-sample and detector-to-sample distance were used to achieve two
nominal voxel sizes of (3.4
CT scanner parameters used for image acquisition of the two carbonate samples. Sample abbreviations are explained in Table 1.
The number of radiographic projections acquired during sample imaging with low and high resolutions were 3001 and 5001, respectively. The total scanning time for one sample was about 8 h. Initial cone-beam 3-D image reconstruction was performed using the internal software XM Reconstructor (XRadia). To remove geometrical artefacts during reconstruction, a secondary reference was acquired for sample images with maximum resolution.
In addition to solid grains and pore space different micritic phases are
visible in the raw images of the scans entailing an advanced segmentation
procedure. For our segmentation (Fig. 2) we select a region of interest
(ROI) from the raw data of the two types of carbonates with two different
resolutions (Table 3). The ROI is subdivided in eight partly overlapping
subvolumes, each of a size of 400
Simplified segmentation workflow as applied in this study.
Our segmentation workflow is applied to the full ROI including all
subvolumes. Image enhancement and segmentation steps were carried out using
the software package Avizo Fire 9 (FEI Visualization Sciences Group). Before
actual segmentation the image noise and scan artefacts are reduced while
preserving interfaces using a 3-D non-local mean image filter. In our
experience the standard values of this filter are appropriate (search
windows
Sketch to illustrate the segmentation geometry (here for Carb-B;
high resolution). The full cube is subdivided into eight partly intersecting
subvolumes with a size of 400
Slices of the raw tiff images of the scanned samples. The dark green areas mark the overlapping zones of the considered subvolumes. Top row: Carb-A with high (left) and low resolution (right). Bottom row: Carb-B with high (left) and low resolution (right).
Colour-coded histograms of the scanned samples. Top row: Carb-A with high (left) and low resolution (right). Bottom row: Carb-B with high (left) and low resolution (right). Between the high-confidence pore phase (marked blue) and the high-confidence mineral phase (marked red) we define five intermediate classes to characterize the micritic phases within carbonate rock.
Slices of the segmented images used for the numerical simulations to determine permeability and velocities. Top row: Carb-A with high (left) and low resolution (right). Bottom row: Carb-B with high (left) and low resolution (right).
The image-enhanced data sets were segmented into classes using global
thresholds for the covered range of grey values. The global threshold is
valid for all the eight partly overlapping subvolumes mentioned above.
Considering that the samples represent quasi-monomineralic calcitic rocks, we
identified the following classes illustrated in Figs. 5 and 6:
high-confidence pores (illustrated with dark blue colour), high-confidence mineral (illustrated with dark red colour), and five intermediate classes.
Note that a non-negligible part of the pore space is below the resolution
limit of the
In order to numerically calculate the effective intrinsic permeability
The intrinsic permeability
To obtain effective
Permeability calculations were realized for subvolumes of the Carb-A and
Carb-B samples. However, the domain size of the Carb-A (0.43 mm) and Carb-B
(0.78 mm) high-resolution samples is smaller than the low-resolution ones
(2.4 mm for Carb-A and Carb-B), i.e. less representative of the material;
therefore we numerically investigate only the extreme porosity
configurations. To select the domains for permeability calculation we adapt
the porosity configurations (range defined by high-confidence pores to
high-confidence grains; see also discussion in Sect. 5.2) showing the
minimum and maximum deviation in porosity with respect to the experimental
investigation. To analyse the homogeneity of the sample this step was
performed for all eight subvolumes. For the high-resolution subdomains (Carb-A
and Carb-B) we perform Stokes flow simulations only in one direction
(
The left-hand side of Fig. 7 displays the intrinsic permeability calculated for the Carb-A high resolution. The porosity range of the subvolumes is higher than the experimentally determined porosity. In addition, the numerically calculated permeability values are significantly lower than the values obtained for the low-resolution samples. The right-hand side of Fig. 7 shows the results of the intrinsic permeability calculated for the selected high-resolution samples of the Carb-B. It can be observed that the high-resolution sample shows a much lower variation between the extreme values of the porosity range.
From the results of the high-resolution samples, Carb-A and Carb-B, it could be observed that the variation in pore channel arrangement is significant and the permeability in the different subvolumes of the same material does not necessarily increase with the porosity increment.
Several micritic phases have been identified in the raw images of the
carbonate rock (i.e. the phases between high-confidence pores and
high-confidence minerals; compare with Sect. 3.2). The porosity of these
regions cannot be determined exactly, as some pores are below the resolution
of the scans: typically, micrites exhibit pore sizes with a maximum diameter
of 1
Intrinsic permeability simulated for the eight subvolumes. Results are given for the extreme porosities configuration of Carb-A (left-hand side) and Carb-B (right-hand side) high-resolution samples.
Effective
Similar to the procedure of the numerical simulation for elasticity
(Sect. 4.1.2),
we vary the sample porosity. This way we get six different porosities
for each subvolume depending on the threshold variation. To reduce
computational times for the Stokes flow simulation we eliminate the
disconnected pores. Some subvolume solid–pore configurations with lower
porosities do not present connected pores, and we assume the effective
permeability
Simulated intrinsic permeability as a function of porosity. Results for the eight subvolumes of Carb-A (left-hand side) and Carb-B (right-hand side) low-resolution samples. Squares markers display each of the pore–solid configurations.
Effective
Left-hand side: intrinsic permeability for subvolume 111 of Carb-A
low resolution. Permeability calculated from flow simulated in
Figure 9 presents the permeability values for Carb-A (left-hand side) and Carb-B (right-hand side) samples as a function of porosity. For the permeability calculations for these samples we perform the Stokes flow simulation in a z-direction only (compare Fig. 3).
Additionally we performed the Stokes flow simulations in three directions
(
From the CT data of the low-resolution Carb-A, the largest domain that could
be extracted is 2.4 mm
For the low-resolution scans we repeat the two-phase simulations for Carb-A
and Carb-B as described in Sect. 4.1.2. The results are displayed in
Fig. 10. Interestingly, the two-phase trend given by Eqs. (4) and (5)
is confirmed only clearly for
Especially in the low-resolution case, we expect to have images with a large number of unresolved porosity, mainly due to micritic phases. Therefore we perform multiphase simulations and vary the porosity by assigning effective elastic properties to an interval of micritic phases (always starting with the class closest to the high-confidence pore phase). As an effective medium approach we use the trend given by the simulations using two single phases only (Eqs. 4 and 5), which is supported by two observations. First, this trend was already observed by Saenger et al. (2014) on different scales on a different carbonate data set. Second, there is an observed self-similarity for carbonates (Jouini et al., 2015). Therefore, despite the interval of micritic phases, we assign vacuum values for the high-confidence pores and use the known elastic moduli of calcite (e.g. Andrä et al., 2013b) for the remaining phases. The results are displayed with green dots for Carb-B in Fig. 10. We repeat the procedure with different intervals of the micritic phases. There are three interesting observations: (1) the resulting effective velocities are always significantly below the observed two-phase trend, (2) the curves for different intervals will intersect each other, and (3) the experimental determined velocities for high confining pressures are between the multiphase results and the two-phase trend as described above.
In this paper we compare results from laboratory investigations with numerical estimates based on digital images. Note that in laboratory experiments we use samples on the centimetre scale for the determination of permeability and ultrasonic velocities and compare it with DRP predictions based on images on the millimetre scale. Especially because of the known heterogeneity of carbonates there is always a risk that the selected scanned area is not representative compared to the full sample size used for laboratory characterizations. In general, a multiscale approach as suggested by Ringstad et al. (2013) should be used for upscaling the results to the plug scale. However, our studies on Carb-A and Carb-B suggest workflows which should be applied in practice for as many samples as possible for improving the statistical significance.
Even with the highest resolution currently available in micro-XRCT imaging there will be a significant amount of unresolved pore features which need to be treated in the DRP workflow (Saenger et al., 2016). On the grey-scale intensity level histograms of the low- and high-resolution images of the micro-CT scanning (Fig. 5), this is reflected in a continuum in the intensity levels between the phase identified as pores and the phase identified as calcite grains. In this paper we have dealt with these micritic phases by replacing, step by step and in a cumulative way, each of the micritic phases by void, and establishing a porosity–velocity trend. A more advanced technique using dry and wet imaging is suggested by Bhattad et al. (2014) using the difference imaging to approximate effective properties. However, the nanoindentation technique provides a measure for the distribution of effective elastic properties at the micrometre scale, and can thus potentially constrain the input parameters for the different phases identified during the segmentation. To be able to do so, nanoindentation needs to provide bulk and shear moduli from each of the measurements (load–displacement curves) and we need to obtain effective bulk and shear moduli values for each of the identified phases in the microtomography (pores, calcite grains, and the five micritic phases). However, if nanoindentation technique is a well-established technique in material sciences, which deals with homogeneous, purely elastic materials, this is, as of today, not the case for rocks, which are heterogeneous materials with both elastic and non-elastic behaviour (creep). Though nanoindentation tests provide significant insights into elastic properties of heterogeneous rocks such as carbonates (Lebedev et al., 2014; Vialle and Lebedev, 2015) or shale (Ulm and Abousleiman, 2006; Abousleiman et al., 2007), there are still some points to be looked at before using the derived values of Young's (or shear and bulk) moduli in a quantitative way for DRP: the value of Poisson's ratio to be used, effect of surface roughness, local mechanical damage induced on the sample's surface by polishing techniques, etc. Nonetheless, the histograms of the indentation moduli of both samples show a broad distribution of moduli values ranging from very low values (a few GPa, where the indenter tip measures stiffness of an area mostly made of a pore) to values consistent with calcite. The existence of these intermediate values is consistent with the existence of micritic phases identified with X-ray tomography. However, we did not observe two peaks in the histogram for the nanoindentation results (Fig. 1) in contrast to the histograms of the scanned micro-XRCT images (Fig. 5). Therefore the direct translation of moduli derived from nanoindentation also remains difficult. The resolution of nanoindentation used in this study allows for determining effective elastic properties at slightly bigger scales than those used here for the micro-XRCT.
Regnet et al. (2014) showed that there is a relationship between micrite microstructure and laboratory ultrasonic velocities on core samples, with samples with a higher content of tight micrite, exhibiting higher velocities, and samples with a higher content of microporous micrite, exhibiting lower velocities. Studied core samples were through a mixture of different types of micrite and the measured velocities represent effective properties at the core scale. This observation is reflected in the established porosity–velocity trends (Eqs. 4, 5, and 6): micritic phases with density closer to that of calcite (tight micrites) have higher velocities than micritic phases with lower density (Figs. 8 and 10), which are closer to that of pores (microporous micrites).
After the segmentation it is also possible to estimate the porosities of the samples. Based on the suggested workflow described in Sect. 3.2 there will be a lower and upper bound. For the lower bound we will treat only the high-confidence pores as pores; for the upper bound we count only high-confidence minerals as minerals.
We obtain a porosity range between 25 and 35 % for the high-resolution data of Carb-A and a range between 7.5 and 31 % for the low-resolution data. We observe that the mean value is in rough agreement with the experimentally determined porosity of 16.7 % (see Table 1) only for the low-resolution case. Although also the experimental value comes with an error we conclude that the high-resolution data set for Carb-A is maybe not representative for the full sample used for the helium porosity in the laboratory. In case of Carb-B the intervals range from 13 to 45 and 7 to 48 % for the high-resolution and low-resolution case, respectively. Here the mean value is in both cases closer to the experimentally determined porosity of 29.4 %.
We conclude that the porosity values of carbonates using micro-XRCT data will only provide estimates with a relatively high uncertainty due to the significant amount of unresolved pore features in the images. An indication is the result of the mercury-intrusion experiments presented by Vialle et al. (2013): the pore throats of the micritic phase are mainly below the resolution of available micro-XRCT devices.
Permeability numerically estimated for Carb-B (Fig. 9, right-hand side) presents an error of 97 % on average with respect to the experimental value. In some cases the error is as low as 55 %. The numerical error in comparison with the experimental values is within the expected range for the numerical method at these porosities, cf. Table 1 in Osorno et al. (2015).
Experimental results for Carb-A sample are below the measurement error tolerance. This could imply a sample with no connected pores between the inlet and outlet defined for the experiment. The numerical estimation of the permeability for the Carb-A low-resolution sample (Fig. 9, left-hand side) is 4 orders of magnitude higher than the experimental measurement (on average 7.0 D). In the high-resolution case (Fig. 7, left-hand side) the numerical estimation is closer to experimental results (on average 0.1 D), but the porosity presents a large numerical error; therefore we do not take this domain as representative of the sample. However the numerically calculated permeability does not differ much from values found in the literature for porous rocks with similar porosity (cf. Andrä et al., 2013b). On the other hand it is observed that for a porosity below 25 %, permeability values of carbonates can span several orders of magnitude (e.g. Fig. 3 of Vialle et al., 2013). Therefore we suggest considering a statistically significant number of samples to characterize a formation and found that eight samples are sufficient for the numerical permeability calculations (see Figs. 7 and 9).
There are two important observations. The two-phase trend (displayed with
solid and dashed/dotted lines) seems to be an upper bound for velocities.
This data-driven upper bound is much stricter than the bound given by
Hashin–Shtrikman (see Fig. 8) and is now confirmed for several carbonates
using several resolutions (this study and Saenger et al., 2014). Only for the
low-resolution images we observe a slightly different trend for
The trend given by the envelope of the multiphase simulations (displayed by dashed/dotted lines in Fig. 10, right-hand side) is not a strict lower bound, because the shape will strongly depend on the applied method to determine effective elastic properties for areas which are below the resolution limit of the used XRCT technique. The best choice to our knowledge is the two-phase trend discussed above, which can be regarded as a carbonate-data-driven effective medium approach. We suggest implementing here in the future also the findings of the nanoindentation experiments. However, we observe that the velocities obtained for the multiphase simulations are in a reasonable agreement with laboratory measurements. This is the case for a known porosity determined in complementary laboratory studies (see also Sect. 5.1). For carbonates the distribution of the micritic phases and their effective elastic behaviour is crucial to predict the effective wave speeds.
In general we do not observe any significant anisotropy for permeability and
for velocities of the considered samples. However, a few samples are out of
this general trend. One example is a subvolume of Carb-A (low-resolution
case), for which we show the results for
With the current imaging techniques it remains difficult to resolve microstructures (on submicrometre scale) and image a representative volume at the same time, which is essential to understand the effective material properties of rocks. For this purpose the exact determination of the porosity of the rock samples is the most relevant parameter. To overcome this problem, we have conducted a specific multiphase segmentation technique and a careful calibration of DRP estimates with laboratory data. Especially for carbonate samples it is difficult to exactly estimate the porosity from raw CT data, because the micritic phases remain unresolved with an unknown porosity. Therefore, we use our presented numerical results in an inverse way. We suggest using the porosity determination from the laboratory and go back to our low-resolution trends given in Figs. 9 and 10. With a given porosity we can now estimate the permeability and the effective wave velocities.
In case of the studied samples Carb-A and Carb-B, we can predict
However, for the used carbonate rock samples anisotropy seems insignificant for elastic as well as for hydraulic properties.
In general, the resolution of the XRCT is the limiting factor for the application of DRP for carbonate rock. The micritic phases remain unresolved even for the highest resolutions available. Therefore, the effective elastic properties have to be approximated. Our suggestion is to use the trend of the two-phase simulations. The implemented workflow in this paper can be applied in general for numerical estimates of mechanical and transport properties of carbonates. Because of the known strong heterogeneity of carbonates we suggest using a statistically significant amount of digital images to characterize a formation.
Erik H. Saenger would like to thank ExxonMobil for the support of some ideas presented in this study. This work was partially funded by Curtin Reservoir Geophysical Consortium (CRGC). The authors thank the National Geosequestration Laboratory (NGL) of Australia for providing access to the X-ray microscope Versa XRM 500 (Zeiss – XRadia) and to the Nanoindentation system (Fisher-Crips Laboratories Pty.Ltd.). Funding for this facility was provided by the Australian Federal Government. Edited by: M. Halisch Reviewed by: O. Lopez and one anonymous referee