基于CT扫描红层砂岩孔隙多标度分形维数的确定方法

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。 

Determining method of multiscale fractal dimension of red bed sandstone pores based on CT scanning

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.