Determining method of multiscale fractal dimension of red bed sandstone pores based on CT scanning
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摘要:
岩石作为一种典型的多孔介质，其内部孔隙结构、形态特征极为复杂，运用常规线性系统内参数描述较为困难，因而采用非线性系统内分形维数这一参数来定量表征孔隙结构的非线性分布特征较为合适。岩石内部孔隙结构分布具有统计意义上的分形特征，因此分形维数的确定对于定量表征孔隙结构分布规律，以及揭示岩石各种力学行为与物理力学指标有着重要意义。将图像处理与阈值分割、分形理论和数理统计相结合，针对CT扫描切片图像，三维重建孔隙结构空间分布模型，计算Hausdorff测度空间下孔隙结构分布盒维数与集束维数。同时，针对孔隙结构空间分布复杂程度的定量表征，提出体素盒维数与圆柱体空间集束维数假想，并通过多种数理统计方法进行假设检验。最后，指出孔隙结构分布是一种多标度分形模型，仅仅单个维数无法描述其全部细节特征，需采用多重分形谱来更为全面表征孔隙分布的细观特征。研究结果表明，Hausdorff测度空间下的平面盒维数可较为全面表征平面孔隙分布特征，但对于针对灰度CT图像构建的体素盒维数可替代传统意义下定义的盒维数，在细观尺度下能更可靠、准确、全面地定量表征孔隙体积分布规律; 集束维数实质是用来定量表征孔隙位置分布规律，若等于欧氏维数，则表明孔隙位置分布具有随机性。
Abstract:The distribution of pore structure inside rock has fractal characteristics in statistical sense, the determination of itsfractal dimension is of great significance to characterize the distribution law of pore structure quantitatively and reveal various mechanical behaviors and physical and mechanical indexes of rock.By combining image processing, fractal theory and mathematical statistics, the spatial distribution model of three-dimensional pore structure was reconstructed based on CT scan slice images, and the distribution box dimension and cluster dimension of pore structure in Hausdorff measure space were calculated. In order to quantitatively characterize the spatial complexity of pore structure distribution, the hypothesis of voxel box dimension and cylinder space bundle dimension was put forward, and the hypothesis was tested by various mathematical statistical methods. Finally, it is pointed out that the pore structure distribution is a multi-scale fractal model, and a single dimension cannot describe all its characteristics.The analysis results show that the voxel box dimension constructed for gray CT images can replace the traditional box dimension, and can quantitatively characterize the pore volume distribution law more reliably, accurately and comprehensively at the meso-scale. In essence, cluster dimension is used to quantitatively characterize the distribution law of pore position. If it is equal to Euclidean dimension, it indicates that pore position distribution has randomness.
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Key words:
- red bed sandstone /
- pore /
- CT scanning /
- voxel box dimension /
- cluster dimension /
- multiscale fractals model
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图 4 CT扫描图像处理流程图
a.原始图像; b.渲染图像; c.孔隙分布平面图; d.重建三维孔隙分布模型; e.孔隙体积分布彩虹梯度模型
Figure 4. CT scan image processing flow chart
图 5 体素盒维数线性(a)与非线性(b)拟合
Figure 5. Linear fitting (a) and nonlinear fitting (b) of voxel box dimension
图 9 红层砂岩内部不同位置平面孔隙分布
Figure 9. Pores distribution in different planar positions of red bed sandstone
表 1 不同体素下孔隙数目
Table 1. Pore numbers with different voxels
nv δv/μm Nδv(F)/个 1 5.50 1 028 723 2 6.93 786 910 3 7.93 673 312 4 8.73 603 525 5 9.40 547 001 6 9.99 504 430 8 11.00 438 079 16 13.86 281 838 27 16.50 180 948 64 22.00 70 507 125 27.50 29 575 216 33.00 14 307 注：δv为等效立方体盒子边长(μm); nv为体素数目(个); Nδv(F)为覆盖孔隙结构所需等效立方体盒子数目(个) 
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表 2 各圆柱体内部孔隙率及孔隙数目
Table 2. Porosity and pore number in different cylinders
圆柱体高h=5 mm 圆柱体底面圆半径r=2.5 mm 圆柱体编号 r/mm 圆柱体体积/mm3 孔隙率/% M(r, h)/个 圆柱体编号 h/mm 圆柱体体积/mm3 孔隙率/% M(r, h)/个 1 0.25 0.98 6.15 17 929 11 0.5 9.82 6.15 169 372 2 0.50 3.93 6.11 69 244 12 1.0 19.64 6.13 330 186 3 0.75 8.84 6.00 154 231 13 1.5 29.46 6.12 497 386 4 1.00 15.71 6.11 273 239 14 2.0 39.28 6.13 664 409 5 1.25 24.54 6.15 424 868 15 2.5 49.10 6.06 834 168 6 1.5 35.34 6.17 607 137 16 3.0 58.92 6.09 1 000 456 7 1.75 48.11 6.09 822 582 17 3.5 68.74 6.17 1 173 874 8 2.00 62.83 6.09 1 072 101 18 4.0 78.56 6.14 1 337 775 9 2.25 79.52 6.19 1 353 305 19 4.5 88.38 6.20 1 503 078 10 2.50 98.17 6.17 1 664 708 20 5.0 98.17 6.17 1 664 708 注：M(r, h)为圆柱体内部孔隙数目
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表 3 不同尺寸圆柱体内孔隙数目
Table 3. Pores numbers in cylinders with different sizes
圆柱体编号 r/mm h/mm 圆柱体体积/mm3 孔隙数目/个 1 0.35 1 0.38 7 063 2 0.70 2 3.08 43 210 3 1.05 3 10.39 127 510 4 1.40 4 24.63 350 012 5 1.75 5 48.10 822 003 6 2.10 6 83.12 1 011 236 7 2.45 7 132.00 2 107 302 8 2.80 8 197.03 2 969 005 9 3.15 9 280.54 4 354 104 10 3.50 10 384.83 6 579 840
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表 4 不同位置平面测量元半径与孔隙数目
Table 4. Measuring element radius and pores number in different planar positions
r/mm M(r)(z=0.5 mm) M(r)(z=1.5 mm) M(r)(z=2.5 mm) M(r)(z=3.5 mm) M(r)(z=4.5 mm) 0.25 55 41 86 68 65 0.5 261 183 348 234 279 0.75 545 404 730 527 631 1 1 011 761 1 280 939 1 128 1.25 1 543 1 222 1 912 1 494 1 703 1.5 2 234 1 715 2 760 2 156 2 373 1.75 3 052 2 392 3 753 2 895 3 262 2 4 007 3 191 4 868 3 798 4 229 2.25 5 130 4 146 6 111 4 821 5 383 2.5 6 358 5 141 7 524 5 920 6 670
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