Vesicle sizes and textures can reflect magma degassing evolution processes, including nucleation, growth, coalescence and other types of mechanical interactions to different degrees (Parmigiani et al., 2017; Pioli et al., 2017; Toramaru, 1995). As natural reservoir space in volcanic reservoirs, the microstructure of vesicles determines the reservoir property. Usually, shapes of volcanic vesicles are investigated using shape parameters such as shape factors and elongation parameters.
In Eq. (6), P is the perimeter and S the area of the 2D pores. In Eq. (7), A, the surface area and V, the volume of the 3D pores. Usually, the closer F is to 1, the rounder the pore shape is, and the smaller the value is, the more complex the 2D pore structure is. FF is different from F. The closer FF is to 1, the closer the pore is to the sphere. The more complex the pore is, the higher the FF value is.
The 2D elongation is denoted as E.
where a and b are the long axis and the short axis, respectively, of each 2D pore fitted as an ellipse. The ellipse fitted has the same area, center of mass and direction as the real pore. The larger E is, the slender the 2D pore is.
The 3D elongation is denoted as EE. Calculated by Avizo Software (Ferreira et al., 2018), EE is the ratio of the eigenvalue in the covariance matrix to the maximum eigenvalue, which can reflect the elongation length of the 3D pore. The more elongated pores generally have smaller EEs.
Table 1 lists the average shape parameters of vesicle group with volume (area) values greater than those of quantile 75%, 50%, 25% and 0% in the 2D and 3D pore spaces. In order to eliminate the specificity of slices, all the pores of 900 slices were selected as the analysis target in 2D characterization.
Parameter Sample Area/volume ≥0% Area/volume ≥25% Area/volume ≥50% Area/volume ≥75% 2D 3D 2D 3D 2D 3D 2D 3D Area/volume fraction (%) W41-1 100.00 100.00 99.61 99.97 97.43 99.87 89.89 98.83 W6 100.00 100.00 99.34 99.96 96.60 99.74 87.07 98.60 G177 100.00 100.00 99.29 99.96 95.87 99.82 83.70 98.98 F/FF W41-1 0.901 5 1.162 2 0.875 3 1.365 2 0.857 3 1.532 9 0.835 9 1.492 2 W6 0.814 1 1.938 5 0.762 2 2.394 6 0.693 1 2.987 4 0.643 8 4.098 4 G177 0.709 4 3.643 4 0.633 3 4.544 8 0.516 5 6.084 7 0.399 7 10.171 0 E/EE W41-1 0.142 5 0.565 4 0.137 2 0.607 3 0.117 2 0.633 1 0.108 3 0.758 4 W6 0.239 0 0.447 6 0.252 8 0.462 2 0.284 4 0.436 1 0.266 9 0.442 8 G177 0.292 6 0.491 5 0.322 0 0.473 7 0.369 0 0.433 3 0.370 5 0.400 3
Table 1. Shape parameters of vesicle groups in 2D and 3D with different partitions
In the 2D slices, the first 25% of the vesicles occupy more than 80% of the area space. While in the 3D vesicle space, the first 25% of the vesicles occupy more than 98% of the volume space. This phenomenon indicates that there is a huge difference in the size of vesicles, and a small number of vesicles occupy the majority of the vesicle space. Certainly, it also reflects the existence of a large number of micropores, which may be invalid or "fuzzy" due to the influence of resolution or noise. The presence of these vesicles greatly limits the characterization effects of the average vesicle shape parameters which change significantly in different vesicle groups (Fig. 5).
Figure 5. Line charts of shape parameters of vesicle groups in 2D and 3D with different quantiles. (a) 3D shape factor FF; (b) 3D elongation factor EE; (c) 2D shape factor F; (d) 2D elongation factor E.
Specifically, the average FF of the first 25% vesicles of W6 sample is higher than that of all the vesicles of G177, but lower than that of the first 75%, 50% and 25% vesicles of G177 sample (Fig. 5a). For 2D slices, the average F of all vesicles of G177 is higher than that of 50% and 25% of vesicles than W6 (Fig. 5c).
The deformation of vesicles is also obvious. The variation of 2D and 3D vesicle elongation of different vesicle groups is disordered. The EE of the first 50% vesicles and the first 25% vesicles of W6 is higher than the first 50% and the first 25% vesicles of G177, but the situation is different in the first 75% and the first 100% pores (Fig. 5b).
In order to accurately characterize the texture of vesicles, it is necessary to select appropriate vesicle group as the target. We selected the 2D vesicles with the first 50% of the area and the 3D vesicles with the first 25% of the volume as the research objectives. These vesicles occupy more than 95% of the total volume (area) (Table 1), which can represent the main characteristics of vesicles.
Among them, Sample W41-1 vesicles have the smoothest edge and the simplest structure, indicating that the volcanic vesicles were formed under slow decompression (Namiki and Manga, 2006), and their shapes were nearly spherical. Vesicles in W6 has a more complex structure and higher elongation ratio than vesicles in W41-1, which indicates that the vesicles underwent shear deformation, coalescence and rupture (Okumura et al., 2016, 2008; Degruyter et al., 2009). In addition, due to the influence of partial filling, FF of G177 was significantly higher than that of others, indicating that the vesicles were broken and the structure became extremely complex.
We also found larger vesicles have lower FF and higher F (Figs. 5a and 5c). It is worth mentioning that the shape factor of W41-1 does not change much, but changes obviously in W6 and G177. Since the larger vesicle bodies of W6 and G177 are mainly of deformation and partial filling vesicles, the difference of pore structure morphology between large and small pores is huge, which leads to significant differences in shape parameters.
In addition, the larger EE in W41-1 3D vesicles indicates that the larger vesicles are closer to the sphere shape with lower elongation ratios. However, the EE of W6 and G177 decreased with the increase of the vesicle volumes, implying that the shear deformation of the larger vesicles is more obvious than that of the small vesicles. This tendency is also shown in the 2D aspect. The first 50% and the first 25% vesicle groups show higher E values (Figs. 5b and 5d).
Figure 6 shows the rose diagram of the first 25% 3D vesicles direction calculated by Avizo software (Table 1). These 25% vesicles can reflect the overall characteristics of the vesicles and avoid the influence of micro pores. The direction of vesicles is determined by two parameters, elevation φ (0-90°) and azimuth θ (0-360°). In this paper, elevation φ is divided into two groups, φ≥45° and φ < 45°.
Figure 6. Rose diagrams of 3D vesicles orientation. (a) φ≥45°, the azimuth θ rose diagram of vesicles in W41-1; (b) φ < 45°, the azimuth θ rose diagram of vesicles in W41-1; (c) φ≥45°, the azimuth θ rose diagram of vesicles in W6; (d) φ < 45°, the azimuth θ plum blossom rose diagram of vesicles in W6; (e) φ≥45°, the azimuth θ rose diagram of vesicles in G177; (f) φ < 45°, the azimuth θ rose diagram of vesicles in G177.
We found that the azimuth θ of vesicles in W41-1 are mostly concentrated at 270°-360° and 90°-180° (Figs. 6a and 6b), which may be due to the influence of pressure and gravity during the nucleation process that the vesicles are slightly ellipsoid. The azimuth θ vesicles in W6 show a trend of convergence towards 180°-270° (Figs. 6c and 6d), indicating that the vesicles began to deform in the shear direction owing to the influence of shear strain, and the anisotropy of vesicles distribution increased. The vesicles in G177 were broken by partial filling, resulting in disordered azimuth θ of the fragments and weakening of anisotropy (Figs. 6e and 6f).
In general, the smaller the box dimension Db is, the smoother the surface of pore and throat is, with weaker complexity and stronger homogeneity, and the better the reservoir performance of rock would be; while Db is closer to 2, that is, the larger Db, the less smooth the surface of pore and throat is, and the more complex the structure of spatial junctions are, the more inhomogeneous the size distribution, which means stronger heterogeneity (Xie et al., 2010; Mandelbrot, 1977). Fractal characteristics of pore structure of 900 samples in each of the three samples are analyzed. The results are presented in Table 2, in which the 2D box dimension, Db, was the average value of 900 slices, and the higher Db represents the relatively more complex microstructure (Xie et al., 2010; Yu, 2006). The average Db show that W6 and G177 had more complex microstructure than W41-1, but there was a small difference between W6 and W41-1, which indicates that shear strain does not cause significant deformation of vesicle microstructure. Figure 7 shows the change in Dbs in 900 slices. In the two-dimensional scale, the vesicular structure of W6 has higher Db than that of W41-1 sample with the same porosity, but the difference is small, which indicates that shear deformation can complicate the vesicle structure, although the effect is limited. The Db of G177 is obviously different from W41-1 and W6, which indicates that mineral filling has a more obvious effect on the microstructure of vesicles than shear deformation in the two-dimensional space.
Sample Db Δα ΔαL ΔαR R W41-1 1.082 3 1.102 1 0.443 0 0.659 1 -0.204 7 W6 1.150 9 1.085 3 0.457 4 0.627 9 -0.161 5 G177 1.476 8 1.073 5 0.416 3 0.657 2 -0.225 4
Table 2. Fractal and multifractal parameters describing the 2D vesicles
The heterogeneous properties of vesicle system reflect the process of degassing and nucleation of vesicles to some extents, and also affect the migration mechanism of vesicular volcanic rock reservoir. To some extent, the heterogeneity of pore distribution in porous media affects the connectivity and permeability of pore structure (Chen et al., 2017). The multifractal method is used to quantitatively characterize the multi-dimensional spatial heterogeneity of volcanic vesicles with different textures, which enhances the characterization of spatial distribution pattern of volcanic vesicles. At the same time, the evolution process of igneous rock vesicles is dynamic, so the study of vesicle heterogeneity can offer new ideas for the analysis of magma degassing evolution and volcanic reservoir prediction.
According to the calculation method of multifractal dimension, the statistical order q of multifractal moment method is selected in the range of [5-5], and the 2D multifractal parameters of the sample vesicle structure are obtained by covering the 2D pore space with boxes of different scales.
Table 2 shows the 2D multifractal parameters, where Δα=ΔαL+ΔαR is the width of multifractal singular spectrum, reflecting the spatial differentiation degree of irregular aggregates under different measures. The larger the Δα of vesicle system is, the more inhomogeneity its distribution is, which often indicates lower permeability (Chen et al., 2017). Since ΔαL and ΔαR, the width of the left and the right part of the multifractal singular spectrum, show the difference between relatively larger and smaller vesicles. And thus R=(ΔαL-ΔαR)/Δα, is used to reflect the asymmetry of multifractal spectrum curve and then to describe the differentiation degree of large vesicles and small vesicles.
It is found that the Δα values of W6 and G177 are lower than that of W41-1, which indicates that the 2D vesicles of W6 and G177 have lower heterogeneity. The asymmetry index R is negative, indicating that vesicles (pores) are differentiated in the slices and micro vesicles (pores) are dominant.
Figures 8 and 9 show the distribution of 2D fractal parameters on different slices. The multifractal spectrum width Δα and the asymmetry index R have great changes on different slices, indicating that there are obvious differences in the characteristics of 2D slices at different positions of the same vesicle system. In addition, the multifractal parameters of G177 have the smallest difference among different slices, while those of W41-1 parameters have the largest variation range and those of W6 are between the two samples.
Table 3 lists the 3D Db of the three samples. Compared with W41-1, W6 and G177 have higher Db, which is due to the deformation, fracture and coalescence of vesicles caused by shear strain. Moreover, partial filling can break the original vesicle space and make the microstructure extremely complex.
Sample Db Δα ΔαL ΔαR R W41-1 2.147 3 2.815 0 1.406 6 1.406 6 -0.000 6 W6 2.427 2 2.265 2 1.039 9 1.216 5 -0.078 3 G177 2.631 8 2.440 6 0.747 2 1.692 3 -0.387 4
Table 3. Fractal and multifractal parameters describing the 3D vesicles
Compared with 2D Db, 3D Db has a higher distinction between W41-1 and W6 (Tables 2 and 3). 2D slices only contain plane information of vesicles structure, while 3D data are the superposition of 2D information, which can more accurately describe the characteristics of vesicle microstructure and show the differences of different vesicle structure.
According to the multifractal analysis method, the plot of f(α) vs. α to represent the 3D multifractal spectrum characteristics of the vesicle structure is obtained (Fig. 10). The multifractal spectrum of the three samples shows typical right partial continuous spectrum distribution pattern, which further reflects the spatial heterogeneity of the samples (Fig. 10). It is obvious that indeed all of the 3D vesicle networks analyzed of the three andesite samples have multifractal geometries.
The Δα value of 3D vesicles of W41-1 sample is higher than that of W6 (Table 3), reflecting that the spatial distribution of W41-1 vesicles is highly heterogeneous, because the process of small vesicles gathering to form large vesicles makes the spatial distribution of vesicles more chaotic, which is consistent with the characteristics of 2D slices. The vesicles in Sample W6 are deformed, elongated and connected due to shear strain, forming a directional arrangement similar to tubular pumice, and the heterogeneity is reduced. Generally speaking, the connectivity and permeability of volcanic vesicles after shear deformation are enhanced (Farquharson et al., 2016). On the other hand, the permeability of vesicles with similar extension direction is better than that of the pore throat system with high structural complexity. (Lai et al., 2018; Kushnir et al., 2017; Pistone et al., 2017; Vona et al., 2016; Degruyter et al., 2009).
The 3D multifractal spectrum width Δα of G177 is between W41-1 and W6, which indicates that shear deformation reduces the heterogeneity of the original vesicle space distribution, while the filling effect makes the vesicles broken, the spatial distribution becomes complex and the heterogeneity is enhanced.
The asymmetry index R of the 3D multifractal spectrum of W6 is slightly lower than that of W41-1, indicating that the smaller vesicles have a higher bulk density. This is due to the formation of micro pores due to the rupture of vesicles under shear strain, and the larger vesicles are easier to gather and connect under shear deformation, which further decreases the density of large vesicles (Okumura et al., 2008). The multifractal spectrum of G177 is strongly right biased, and correspondingly the asymmetry index R reaches -0.387 4, indicating well that the differentiation degree of larger vesicles and small vesicles is relatively smaller.
We found that the 2D and 3D Δα can effectively characterize the heterogeneity of W41-1 and W6. For the complexly reconstructed vesicle space such as G177, due to the limited information contained in the 2D slices, the shape and spatial distribution of the vesicles are ignored, which makes the 2D heterogeneity and 3D heterogeneity different (Tables 2 and 3).