Introduction

Detailed parametrization of the pore space in tight, gas-bearing, clastic rocks is a key in understanding the porosity distribution and processes of the fluid flow through the pore space, especially important in hydrocarbons exploration (Karpyn et al. 2009; Zhang et al. 2017; Ghanizadeh et al. 2017; Krakowska 2017; Liu et al. 2018). The paper presents the combination of various laboratory measurement results: computed X-ray tomography (nanoscale), nuclear magnetic resonance spectroscopy and also pulse- and pressure-decay permeability method. Computed X-ray tomography (CT) is a well-known method in medicine and material engineering (Al-Raoush and Papadopoulos 2010; Kaczmarek et al. 2017). Physical bases of the method are connected with the X-ray absorption coefficient (Stock 2008). Pore space parametrization was based on the computed X-ray tomography and brought many useful information regarding the pore volume, pore shape and size, as well as the skeleton analysis. Geometrical parameters from CT correspond to the NMR results in the form of T2 relaxation time parameters. The ability of fluid flow through the tight pore space is strongly controlled by the pore sizes, shapes and distributions. However, the absolute permeability can be estimated based only on the geometrical parameters of the pores and microcracks using advanced statistical tools as multiple linear regression analysis.

Computed X-ray tomography has an unquestionable advantage because it does not destroy the material and shows the selected objects, pores or minerals, in the 3D aspect (Wellington and Vinegar 1987; Kayser et al. 2006; Jarzyna et al. 2016; Krakowska et al. 2016; Puskarczyk et al. 2018). Still, it is many to research regarding the tight, gas-bearing formations, which are the object of the interest of petroleum industry (Josh et al. 2012; Guo et al. 2015; Verri et al. 2017). Hence, computed X-ray nanotomography is a valuable laboratory method for the qualitative and quantitative analysis of complicated pore structures and permeability estimation (Cnudde and Boone 2013; Mostaghimi et al. 2013; Krakowska and Puskarczyk 2015; Sun et al. 2017).

Materials and methods

Research material consisted of Paleozoic 20 tight sandstone and 11 mudstone samples, from deep wells located in Poland. Sandstones come from Peri-Baltic syneclise, Pomeranian anticlinorium, Holy Cross anticlinorium and Lublin synclinorium, while mudstones Peri-Baltic syneclise and Lublin synclinorium.

Several laboratory methods were carried out on core samples: computed X-ray tomography (CT) in the form of X-ray nanotomography, nuclear magnetic resonance spectroscopy (NMR), as well as pulse- and pressure-decay permeability method (PDP). CT provides 2D and 3D images of material for qualitative and quantitative analysis of selected objects: minerals or pores (Ketcham and Carlson 2001; Caubit et al. 2009; Cnudde et al. 2011). Highly specialized algorithms are necessary to obtain the proper visualization and quantitative information from objects. Firstly, CT data were processed (Feldkamp et al. 1984; Jędrychowski et al. 2017) and next analyzed in newly developed software poROSE (Habrat et al. 2017; Krakowska et al. 2018), in which many geometrical parameters were implemented for the proper 3D analysis of porous materials. poROSE software contains algorithms, which allow to parametrize the selected objects, e.g., pores, microcracks or minerals, and were built upon basis of the knowledge from different disciplines, as material engineering, medicine, petroleum geology and petrophysics (Madejski et al. 2018).

The following morphological parameters were taken into consideration in pore space characterization: thickness mean, equivalent diameter, anisotropy, elongation, sphericity, Feret diameter, Feret coefficient (determines the ratio of the maximum diameter of the object measured in two perpendicular directions), Feret shape (the ratio of the maximum length of the Feret diameter measured in the direction perpendicular to the line defined by the shortest Feret diameter to the length of the shortest Feret diameter), shape factors: 2nd circularity coefficient (determines the diameter of the circle with a perimeter length equal to the length of the perimeter of the tested object), Malinowska coefficient (combination of objects perimeter and surface area) and Danielsson coefficient (combination of object surface area and minimum distance to the object contour); from the skeleton analysis: junctions, isolated objects, branches, end branches, coordination number; as well as the result of spherical and ellipsoidal pore analysis. Calculations referred to the objects: pores and microfractures. Nuclear magnetic resonance spectroscopy (NMR) was carried out on saturated samples and processed using individual T2 cutoffs. Moreover, pulse- and pressure-decay permeability methods were conducted for proper absolute permeability estimation (Handwerger et al. 2011). Detailed description of tools is presented in Table 1. The goal of the analysis was also to combine parameters from the CT and NMR, PDP, so an effort was made to search for the relationships between the different parameters.

Table 1 Laboratory tools description
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Table 2 consists of the quantitative geometrical parameters of pores and microcracks in the form of median value calculated in the each samples (Feret diameter, sphericity, Feret shape, Feret coefficient) as well as total porosity values and grain size class. Although the samples come from different geological formations and wells (Table 2), they have a common feature in the form of low porosity (0.02–4.42%) and maximum Feret diameter. Pores are characterized with the significant deviation from the shape of the ball, more elongated and edgy.

Table 2 Research material characteristic. Values of Feret diameter, sphericity, Feret shape, Feret coefficient calculated for pores and microcracks are presented in the form of median value for the sample
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Results and discussion

All 31 samples were analyzed qualitatively and quantitatively. Exemplary visualization of the Cambrian tight sandstone is presented in Fig. 1, while Silurian mudstone sample is in Fig. 2.

Fig. 1
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3D image of the tight sandstone pore space (core sample from Cambrian, sample 17), image size: 700 × 700 × 2000 voxels (350 × 350 × 1000 µm3), poROSE software

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Fig. 2
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3D image of the mudstone pore space (core sample from Silurian, sample 26), image size: 500 × 500 × 1880 voxels (250 × 250 × 940 µm3), poROSE software

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Pore size distribution for exemplary samples of sandstone and mudstone is illustrated in Figs. 3 and 4 in the form of equivalent diameter and thickness mean parameters. Tight sandstones consist of pores within large range of diameters in comparison with mudstones. Most of the pore diameters are below 10 µm in the mudstone group, while in the sandstones below 50 µm. Analysis provided information about the number of objects in each sample. It appeared that mudstones are represented by the larger group but with smaller diameters. It is worth paying attention that the sizes of compared samples (Figs. 3 and 4) were different. Sample 17 (Fig. 3) has a voxel size 700 × 700 × 2000, while sample 26 (Fig. 4)—500 × 500 × 1880.

Fig. 3
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Pore size distribution of exemplary tight sandstone sample 17 in the form of equivalent diameter and thickness mean in micrometers

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Fig. 4
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Pore size distribution of exemplary mudstone sample 26 in the form of equivalent diameter and thickness mean in micrometers

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Shape factors give an insight into the pore shape, which influences the filtration abilities. Danielsson coefficient is very sensitive for the pore shape (low values for the elongated objects), while Malinowska coefficient is more likely sensitive for the pyramid shape. Slight variability with the size and the rotation shows Danielsson coefficient, while Malinowska coefficient is stable. Shape factors were implemented and calculated using adopted 3D algorithms. Figures 5 and 6 show the Danielsson coefficient in the sandstone and mudstone group, for each sample, in the form of median and quartile values. Pores in tight sandstones have more complicated shapes and differed within the group. Mudstone pores are more similar in shape. Samples 30 and 31 were probed from the wells in the Peri-Baltic syneclise, while the rest of the samples—from the Lublin synclinorium, hence the difference in the Danielsson coefficient.

Fig. 5
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Danielsson coefficient for the tight sandstone samples

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Fig. 6
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Danielsson coefficient for the mudstone samples

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Shape factors were combined together with the NMR results. Several data sets were prepared with the different statistics calculated for each parameter in each sample, as median, percentile, quartile, for all detected objects in the sample (pores and microcracks). Malinowska coefficient correlates with T2 cutoff 1 for clay bound water and capillary bound water (R = 0.61), T2 cutoff 2 for bulk water irreducible and moveable fluid volume (R = 0.63) for the 10 percentile values, as well as with T2 cutoff 2 for bulk water irreducible and moveable fluid volume (R = 0.60) for the median values. Second circularity coefficient (R = 0.61) and Danielsson coefficient (R = 0.61) built the relationships with T2 cutoff 2 for bulk water irreducible and moveable fluid volume for the 10 percentile value data set.

Moreover, spherical and ellipsoidal pore shape analysis was carried out regarding the pore shape. Spherical pores parameter describes number of spherical pores per unit of rock volume (unit 1/m3 or 1/voxels). Average parameter of spherical pores in sandstone group is lower than in mudstones and is equal to 1.78E − 7 and 5.84E − 7, respectively. It means that more spherical pores are detected in mudstones than in sandstones, in the rock volume. In comparison, sandstone samples 5 and 6 have about 300 spherical pores, while samples 15 and 16 less are than 35. Mudstone samples are characterized by higher number of spherical pores in comparison with sandstone samples, which is around 300 spherical pores. Average diameter of spherical pores is quite similar, more or less about 1.05 µm for sandstones, with 0.24 µm of standard deviation, while for mudstones 0.96 µm and 0.20 µm, respectively.

Ellipsoidal pores parameter is equal to 1.90E − 6 for sandstones and 4.91E − 6 for mudstones and is higher in mudstones. It means that relatively there are more spherical and ellipsoidal pores in mudstones, but there is no information about the pore volume. Sandstone samples have in average about 1600 ellipsoidal pores, while mudstones—2700. Sample volumes were not the same, what influences the number of objects. Nevertheless, in general more objects were detected in mudstone samples. Pore shape factor is ratio of shorter radii of the axis to the longest, evaluated in each plane. Pore concentration factor is calculated by the product of the number of pores per unit of rock volume and the length of the longer half-axis of the ellipsoid. Pore shape factor is around 0.52 and pore concentration factor around 6.1 for mudstones and 0.48 and 2.38 for sandstones, respectively. Moreover, pore concentration factor is very diverse, which means that samples differ with the longer half-axis.

3D skeleton analysis includes information about junctions (Junctions Pxs Count; pixels that intersect on at least two branches), isolated objects (Isolated Pxs Count), ends of branches (End Pxs Count), branches (pixels that create run and end with junction). It provided unique information about the pore space connections and complexity. Results of 3D skeleton analysis in the form of junction and branch visualization are presented in Figs. 7 and 8. Detailed quantitative results are illustrated in Table 3.

Fig. 7
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Image of the skeleton analysis for the Cambrian tight sandstone sample 17

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Fig. 8
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Image of the skeleton analysis for the Silurian mudstone sample 26

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Table 3 Results of basic skeleton analysis for each sample. Symbols as in Table 2
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Detailed skeleton analysis provided information about the average coordination number (Ridgway & Tarbuck 1967). Average coordination number in many cases was a negative number. That is example of poorly developed pore space. Only 2 from 11 samples from the mudstones group had a positive value and only 10 from 20 samples in sandstone group, what indicated that sandstones are characterized by more complex structure. Number of junctions varied from 6 to about 31,686, for arithmetic average 3701 in sandstones, while in mudstones from 31 to 2210, with average 668. It means that sandstones are characterized by pores with complicated shapes, so after transformation into skeleton resulted in higher number of Junctions. However, the number of junctions and branches is not very high, as for conventional rocks.

Different definition of coordination number is connected with the number or branches which are connected with the single pore (Rabbani et al. 2016). Coordination number defines quality of pores connection. Maximum value of coordination number in sandstone group is 17, while average is equal to about 3 (Table 4). Mudstones are characterized with maximum coordination number equal to 6 and average about 3. It indicates that sandstones are more diverse in pores connection in comparison with mudstones but on average they are very similar. Almost all samples have low number of junctions, with the highest average for the Cambrian sandstone 7 and Silurian mudstone 30.

Table 4 Coordination number (CN) for all samples
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Maximum coordination number in the sample (Max CN Sample) correlates with T2 cutoff 2 for bulk water irreducible and moveable fluid volume (R = 0.63) and logarithmic T2 mean T2ML (R = 0.63). There was no relationship detected for the coordination number and absolute permeability.

It is worth paying attention to the strong relationships between following parameters from different laboratory methods: logarithmic mean of transverse relaxation time from NMR and number of junctions from CT, logarithm of absolute permeability from PDP and anisotropy from CT, T2 cutoff 2 from NMR (for bulk water irreducible and moveable fluid volume) and elongation from CT. Figures 9, 10 and 11 show the mentioned relationships. Parameters from X-ray nanotomography usually build the relationships with the NMR spectroscopy results in the form of T2 relaxation time because it corresponds to the pore sizes. Mudstone samples in Fig. 9 concentrate in the range of low logarithmic T2 mean (T2ML) from NMR and low Junctions Pxs Count from CT, revealing poorer development of the pore space. Figure 10 shows dependence of logarithm of absolute permeability from pulse- and pressure-decay method on anisotropy from CT. Sandstone samples are divided into two groups: group of low absolute permeability and high anisotropy values from CT and group of high absolute permeability and low anisotropy values. Mudstone samples cover almost all range of anisotropy values and lower values of absolute permeability. Dependence of T2 cutoff 2 (for bulk water irreducible and moveable fluid volume) from NMR on elongation from CT is presented in Fig. 11. Two groups of sandstone samples are visible: group of higher elongation values from CT and lower T2 cutoff 2 and group of lower elongation values and higher T2 cutoff 2. Mudstone samples are characterized by the higher elongation values, except two samples, which have more pores with sphere shape.

Fig. 9
figure 9

Dependence of logarithmic T2 mean (T2ML) from NMR on Junctions Pxs Count from CT. Colors: blue—sandstones, green—mudstones

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Fig. 10
figure 10

Dependence of logarithm of absolute permeability from pulse- and pressure-decay method on anisotropy from CT. Colors as in Fig. 9

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Fig. 11
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Dependence of T2 cutoff 2 (for bulk water irreducible and moveable fluid volume) from NMR on elongation from CT. Colors as in Fig. 9

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Multiple linear regression method (MLR) was applied to search for the best equation for the absolute permeability estimation using only X-ray nanotomography results (TIBCO 2017). Analysis was conducted in Statistica 12. The purpose is to establish the formula for minimizing the core destruction, as a very valuable material. MLR was done for the all clastic samples taking into consideration the one dependent parameter—logarithm of absolute permeability from pulse- and pressure-decay permeability method and independent variables as all available geometrical parameters from CT. The number of independent variables is estimated based on the total number of samples. In this case, on one dependent variable there are three independent variables for 31 samples. Obtained formula for the logarithm of absolute permeability allows to determine the absolute permeability with correlation coefficient equal to R = 0.78, while determination coefficient R2 = 0.61. There is no strong dependence of the MLR absolute permeability from period, only one for lithology—for sandstones. MLR absolute permeability correlates with PDP permeability for sandstones. Although, mudstones reduced the correlation, it is possible to estimate the absolute permeability also for this group using the estimated formula.

The formula has the form of Eq. (1):

$$\log K = - 7.501 + \left( { - 39.3116*{\text{Anisotropy}}} \right) + \left( {13.6020*{\text{Feret Shape}}} \right) + ( - 5.0714*{\text{Feret Coefficient}})$$
(1)

Table 5 presents the partial regression coefficient and the standardized one. Pore anisotropy has the largest influence on the absolute permeability, so the pores ability to fluid flow.

Table 5 Multiple linear regression results for absolute permeability estimation
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Figure 12 illustrates the comparison between the logarithm of absolute permeability form PDP (log K) and estimated based on MLR (log K MLR).

Fig. 12
figure 12

Comparison of absolute permeability logarithm from PDP (log K) and estimated from MLR (log K MLR). Colors as in Fig. 9

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Conclusions

X-ray nanotomography allowed for the complex parametrization of the pore space in the tight clastic rocks. Selected parameters connected with the geometrical features of pore space can be referred to the ability of fluid flow, as anisotropy, Feret shape or Feret coefficient. Shape factors, as Danielsson, Malinowska coefficient, 2nd circularity coefficient build relationships with the NMR spectroscopy parameters, which is related to the pore sizes. 3D skeleton analysis revealed the information about the pore structure, which in analyzed case is strongly influenced by the compaction and mineralogy. Pores are more elongated, angular, what definitely is not a benefit. Skeleton analysis provided the information about poor quality of the pore connections in the form of coordination number in both, sandstone and mudstone group. Dependence of logarithmic T2 mean (T2ML) from NMR on Junctions Pxs Count from CT, as well as T2 cutoff (for bulk water irreducible and moveable fluid volume) from NMR on elongation from CT, was observed for all the samples.

Using result from pulse- and pressure-decay permeability method, it was possible to estimate the equation for absolute permeability having only data from X-ray nanotomography image analysis. It is an advantage in the times of balance between costs of coring and laboratory measurements. Multiple linear regression was a key in determining the formula. Permeability in tight, gas-bearing clastic rocks is influenced by the anisotropy of pore shape and size. Presented equation can be used in the initial assessment of the reservoir potential of tight clastic rocks. Computed X-ray tomography together with other laboratory techniques, as nuclear magnetic resonance spectroscopy, is a valuable source of detailed parametrization of the pore space in rock samples.