Fractal and Multifractal Characteristics of Pore Throats in the Bakken Shale
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Abstract
To evaluate pore structures of the Bakken Shale, which is one of the most important factors that affect petrophysical properties, high-pressure mercury intrusion was employed in this study. Pore structures such as pore-throat size, pore-throat ratio, and fractal attributes are investigated in this major shale play. Pore-throat size from 3.6 to 200 um is widely distributed in these shale samples. Accordingly, pore-throat size distributions demonstrate the multimodal behavior within the samples. The whole pore-throat network can be divided into four clusters: one set of large pores, two transitional/intermediate pore groups, and one set of smaller pores. The fractal analysis revealed that fractal dimensions decrease as the pore-throat size decreases. The multifractal analysis demonstrated that as the maturity of the shale samples increases, pore-throat size distributions would become more uniform and pore structures tend to become more homogeneous. The results are compared to our previous results obtained from nitrogen gas adsorption for further verifications of fractal behavior. Finally, although fractal analysis of mercury intrusion and nitrogen gas adsorption were comparable, the results of multifractal analysis from these two methods were not identical.
Keywords
Shale Pore-throat size Mercury intrusion Fractal Multifractal1 Introduction
Unconventional shale formations (including shale oil and shale gas) have become a major source of hydrocarbon production in recent years (Hu et al. 2017). Shale is a fine-grained, clastic sedimentary rock which is usually composed of a mixture of clay and other minerals such as quartz, feldspar, and calcite (Blatt et al. 2006). Compared with conventional plays, shale reservoirs have lower porosity and permeability which makes them to categorized under unconventional reservoir (Liu et al. 2017a; Zhang et al. 2017; Li et al. 2017a). Storage and flow of hydrocarbons through such formations is controlled by the capillary entry pressure, the permeability and the extent of the diffusive losses through the pore spaces (Schlömer and Krooss 1997; Schmitt et al. 2013; Yang et al. 2016). Therefore, a better understanding of pore structures/network of these shale formations can assist in evaluating the overall reservoir production performance.
High-pressure mercury intrusion is an extremely useful characterization technique for porous materials and is considered as one of the few methods that can acquire data over a broad dynamic range of pore sizes using a single theoretical model (Webb 2001; Giesche 2006). With the strong characterization capabilities, high-pressure mercury intrusion can measure capillary diameter in ranges from 3.6 nm to 360 um which is considered a wide interval of pore sizes in porous rocks. This is far beyond what nitrogen gas adsorption can detect which is limited to pore sizes less than 200 nm. High-pressure mercury intrusion has been widely applied in characterizing pore structures of coal (Peng et al. 2017; Zhou et al. 2017; Yu et al. 2018), carbonate (Ding et al. 2017), shale oil reservoir (Hu et al. 2017), shale gas reservoir (Schmitt et al. 2013; Labani et al. 2013) and tight oil sandstone (Li et al. 2017b).
Other researchers and we have shown that micropore structures of shale formations are very complex with a wide range of pore sizes from nano to macro-meters in diameter (Chen and Xiao 2014; Liu et al. 2017a, b, 2018a). These complex pore structures sometimes can’t be modeled by traditional Euclidean geometry (Lopes and Betrouni 2009). In this case, fractal theory which is introduced by Mandelbrot (1983) has now become a powerful tool to characterize pore size distributions (Liu et al. 2017a; Liu and Ostadhassan 2017; Xia et al. 2018) and electrical conductivity (Cai et al. 2017) of shale formations. According to the theory of fractal geometry, the fractal dimension of the surface varies between 2 and 3. If the fractal dimension value is close to 3, it indicates that the pore structures are very complex (Thompson et al. 1987). However, the single fractal dimension can only represent the complexity of pore structures and suffers from quantifying heterogeneities of overall pore network of shale rocks. The latest one can be studied via multifractal analysis (Liu et al. 2018b). Multifractal analysis is the extension of fractal analysis which uses a set of generalized dimensions instead of the single dimension, D, common in fractal analysis. Multifractal analysis has been applied in the study of pore structures of mercury intrusion data from soil (Paz Ferreiro et al. 2010) and coal samples (Yu et al. 2018) and has shown great potential in providing us with in-depth information regarding pore structures compared to single fractal theory.
In previous attempts, we applied both fractal and multifractal methods to analyze pore structures of the Bakken Shale samples on the data that was collected by gas adsorption (Liu et al. 2018b). In this study, high-pressure mercury intrusion capillary pressure data is used to characterize pore structures of the Bakken shale. The purpose of this study is to address the following questions: (1) What are the pore-throat characteristics of the Bakken shale? (2) What is the fractal dimension and heterogeneity of the Bakken shale pore network using high-pressure mercury intrusion method? (3) Is there any correlation and meaningful relationship between fractal information from high-pressure mercury intrusion and low temperature nitrogen adsorption?
2 Experiments and Modeling
The schematic of the whole process will include the following steps: first, we chose the samples and derived the basic properties of the samples (mineral compositions and geochemistry properties); Then, we calculated the pore-throat distributions and analyzed the characteristics of the pore throats of the samples from high-pressure mercury injection; After that, we quantified the fractal behavior and the multifractal behavior of the pore-throat distributions. Finally, we compared the fractal behaviors and multifractal behaviors and found their correlations.
2.1 Samples and Experiments
Five samples from the Bakken Formation were collected and then analyzed with X-ray diffraction (XRD) for mineralogical compositions along with Rock–Eval pyrolysis to quantify total organic carbon (TOC) content and thermal maturity (Liu et al. 2018a). For the high-pressure mercury intrusion, all the samples were first vacuum-dried at 70 °C in the oven for more than 10 h and then were moved to the mercury porosimeter (Auto Pore IV 9510, Micrometrics Instrument). The injection pressure was increased from 0 up to 60,000 psi (413.68 MPa) to obtain the relevant capillary pressure versus mercury saturation data.
2.2 Single Fractal Analysis
2.2.1 Geometric Fractal Dimension
This will create a linear relationship between, log(1 − S_{g}) and logP_{c}. Hence, the fractal dimension can be directly calculated from the slope of this cross-plot.
2.2.2 Thermal Dynamic Fractal Dimension
2.3 Multifractal Analysis
The detailed multifractal analysis procedure can be found in our previous studies which are focused on the gas adsorption data analysis (Liu et al. 2018b). Briefly, a set of boxes with equal length (ε) are applied to the pore size distribution data.
3 Results and Discussions
3.1 Sample Compositions Analysis
Mineral compositions and maturity of the samples
Samples | Quartz (wt%) | Pyrite (wt%) | Feldspar (wt%) | Dolomite (wt%) | Clays (wt%) | VRo-Eq (%) |
---|---|---|---|---|---|---|
Sample 1 | 70.30 | 3.15 | 7.70 | 0.00 | 18.81 | 0.56 |
Sample 2 | 54.30 | 8.07 | 0.00 | 8.80 | 28.60 | 0.56 |
Sample 3 | 66.90 | 2.44 | 14.40 | 0.00 | 16.20 | 0.63 |
Sample 4 | 70.00 | 2.35 | 5.50 | 0.00 | 22.20 | 0.63 |
Sample 5 | 36.10 | 8.50 | 3.40 | 0.00 | 52.00 | 0.92 |
3.2 MICP Curve Analysis
For the extrusion part, the saturation value of all the samples will decrease as the extrusion pressure decreases. Hysteresis between the intrusion and the extrusion is observed for all samples. If all pores are ideally uniform and cylindrical in shape and the intrusion and extrusion of the contact angles are known, then hysteresis could not be expected since the intrusion and the extrusion process are controlled by the same mechanism and exact known parameters (Webb 2001). However, in reality, most samples do not have the ideal pore geometry. As the mercury retracts from the pore system, the new mercury interfaces will be created, and additional energy is needed to get mercury out of the pores. During the intrusion process, a pore is filled with mercury not only due to the pore size being equal or larger than the corresponding pressure but also because of a continuous path that mercury needs to follow to get to that specific pore. The large internal pores which are surrounded by smaller ones can only get filled until the pressure is sufficient to fill and follow a pathway toward that pore completely. During the extrusion process, the reverse phenomenon occurs. Those filled internal pores or isolated pores will remain filled with the trapped mercury if they do not own a continuous path toward the sample surface for mercury to leave them (Giesche 2006). The mercury withdrawal efficiency of these samples was calculated from 24.25 (Sample 4) to 37.61% (Sample 2), indicating the complex pore network of these shales.
3.3 Pore-Throat Size Distribution from Mercury Intrusion Analysis
Overall, as the maturity increases, the percentage of the ratio of larger pores over total pore volume of these shale samples will increase. A closer look at Fig. 3 confirms even samples at almost the same maturity level and would exhibit dissimilar pore structures. Considering Sample 1 and Sample 2 for example, only 6% of the total pore volume is made of pore throat with size less than 10 nm in Sample 1 and less than 10 nm comprised almost 13.38% of the total pore volume of Sample 2. The largest pore-throat ratio of Sample 1 was found 66.1, while for Sample 2 the largest pore-throat ratio is 174.7. All these differences indicate that thermal maturity is not only the sole governing factor on pore structure characteristics but also some other controlling components such as mineral compositions could play an important role on various attributes of pore structures. Pore-throat ratio decreases as the pore-throat size decreases which is consistent with the results found by other researchers (Hu et al. 2017).
3.4 Fractal Analysis
Fractal analysis of the samples
Large pore | Transitional pore 1 | Transitional pore 2 | Small pore | |||||
---|---|---|---|---|---|---|---|---|
D _{ l} | R ^{2} | D _{ t1} | R ^{2} | D _{ t2} | R ^{2} | Ds | R ^{2} | |
Sample 1 | 2.9753 | 0.9969 | 2.9547 | 0.8903 | 2.3643 | 0.9949 | 2.0953 | 0.9884 |
Sample 2 | 2.9633 | 0.9996 | 2.9132 | 0.8492 | 2.5378 | 0.9976 | 2.1627 | 0.9789 |
Sample 3 | 2.9709 | 0.9942 | 2.7369 | 0.9977 | 2.5171 | 0.9984 | 2.0399 | 0.9813 |
Sample 4 | 2.9716 | 0.9966 | 2.8198 | 0.9352 | 2.5246 | 0.9937 | 2.1449 | 0.9964 |
Sample 5 | 2.9160 | 0.9904 | 2.8485 | 0.9920 | 2.4868 | 0.9883 | 2.0933 | 0.9858 |
3.5 Multifractal Analysis
The generalized dimension spectrum of all the samples
D _{−10} | D _{0} | D _{1} | H | D _{10} | D_{−10}–D_{0} | D_{0}–D_{10} | D_{−10}–D_{10} | |
---|---|---|---|---|---|---|---|---|
Sample 1 | 1.6712 | 0.9997 | 0.2287 | 0.5208 | 0.0442 | 0.6715 | 0.9555 | 1.627 |
Sample 2 | 2.1137 | 0.9997 | 0.2456 | 0.5218 | 0.047 | 1.114 | 0.9527 | 2.0667 |
Sample 3 | 1.6499 | 0.9997 | 0.2871 | 0.5264 | 0.0573 | 0.6502 | 0.9424 | 1.5926 |
Sample 4 | 1.6122 | 0.9997 | 0.2476 | 0.5217 | 0.0469 | 0.6125 | 0.9528 | 1.5653 |
Sample 5 | 1.6073 | 0.9997 | 0.4627 | 0.5606 | 0.1208 | 0.6076 | 0.8789 | 1.4865 |
The width of (D_{−10}–D_{10}) can be applied to indicate the heterogeneity degree of pore size distribution. The larger (D_{−10}–D_{10}) value reflects more heterogeneity within the pore size distribution. The average (D_{−10}–D_{10}) value of Sample 1 and Sample 2 is around 1.84685 which is larger than the average value (of 1.57895) of Sample 3 and Sample 4. Sample 5 which is the most mature sample among all has the smallest (D_{−10}–D_{10}) value. As the maturity increases, the pore size distribution will become more homogeneous. This can be interpreted as the thermal advance in the samples will lead to an evolution of larger pores within the organic matter which has been reported in several articles (Chen and Xiao 2014; Liu et al. 2017a, b).
3.6 Correlations Between the Fractal Analysis and Multifractal Analysis
3.7 Comparison the Fractal Information from Mercury Intrusion and Nitrogen Gas Adsorption
4 Conclusions
- 1.
Hysteresis exists between intrusion and extrusion curves, indicating a complex pore structures in the samples. This infers that large pores are interconnected by smaller pore throats.
- 2.
As the maturity increases, the percentage of large pores to the total pore volume will increase referring to porosity evolution within the organic matter. Pore-throat ratio decreases as the pore-throat size increases. A few inflection points were found in the pore size distributions and the location of the inflection point varies with the samples.
- 3.
The fractal analysis shows that mercury intrusion curve can be subdivided into four segments and the fractal dimension of these four segments follows the order of: D_{l} > D_{t1} > D_{t2} > D_{s}. The fractal dimension of large pores is largest, while the fractal dimension of the small pores is smallest.
- 4.
The pore structures investigated from mercury intrusion show a multifractal behavior of the samples. The pore size distribution is becoming more uniform as the shale samples are getting more mature. There was not any correlation between fractal dimensions (D_{w} and D_{a}) and heterogeneity value (D_{−10}–D_{10}). Larger pores make a significant contribution to multifractal parameters when q > 0, while the small pores and transitional pores are the main contributors to the multifractal parameters when q < 0.
- 5.
Multifractal analysis of nitrogen gas adsorption and mercury intrusion were compared, and a major discrepancy was pointed out between the results of each, originating from the heterogeneity of the pore structures.
Notes
Acknowledgements
The authors appreciate the support from China Scholarship Council (No. 201406450029). We would like to also show our appreciation to ND Core Library, Jeff Bader the director and state geologist as well as Kent Holland library technician for providing us with the samples. We thank Dr. Liu from Northeast Petroleum University for running the experiments. We also appreciate the reviewers to give their comments to improve the quality of the paper.
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