低渗煤岩气液两相流分形运动方程

Fractal Dynamics of Gas-liquid Flow in Low-permeability Coal

煤岩等多孔介质中毛细管气液两相流规律是解决渗透率理论表达的认识基础,也是揭示低渗机制必经路径。基于微纳米尺度上的毛细管分形结构,具体为纵向上建立分形迂曲度模型与横向上建立分形截面模型,定量解析低渗煤岩孔隙结构。分析了基于Hagen–Poiseuille方程与毛管压力公式的迂曲度定义差异,后者结合压汞实验更适用于微米尺度以下孔道渗流描述。基于分形标度关系,分析了长径比、迂曲度分维对分形影响系数的影响。开展了煤岩压汞试验,计算得到平均迂曲度与分维,并定义二者乘积为迂曲度系数。基于Carman–Kozeny方程推导了毛细管截面分形结构方程,包括周长分维与面积分维理论表达式。建立了气液两相驱替模型,基于Washburn运动方程,推导了界面位置与速度的分形运动方程。基于Nano–CT重构了低渗煤岩纳米-微米孔隙结构,获得了孔径与孔隙体积分布,结合分形模型计算了面积分维与周长分维。最后基于NMR实验开展了两相N2–H2O驱替实验,获得了实验驱替下界面运动距离与速度分布。结果表明:毛细管分形结构可作为解释低渗机制的几何桥梁。迂曲度系数可全面反映迂曲度与迂曲分维的影响,近似与渗透率呈线性关系。验证了分形截面模型的可靠性,指出煤岩低渗机制尚应考虑分形截面粗糙度系数的影响。通过驱水信号分布证实了分形运动方程的有效性。

The gas-liquid flow in the porous medium of coal is the foundation for understanding the theoretical expression of permeability, and it is also play a vital role in revealing the mechanism of low-permeability. Based on the fractal structure of capillary in micro and nano scales, a fractal tortuosity model established in the longitudinal direction and a fractal cross-section model established in the transverse direction were employed to quantitatively analyze the pore structure of low permeability coal. The differences in the definition of the tortuosity based on the Hagen–Poiseuille equation and the capillary pressure formula are analyzed. The latter one combined with the mercury intrusion experiments is more suitable for the seepage of coal pores below the micro scale. Based on the relationship of fractal scales, the effects of the aspect ratio and the fractal dimension of tortuosity on the fractal coefficient were theoretically analyzed. The mercury intrusion tests in coal were also carried out, and the average tortuosity and fractal dimensions were calculated; the product of the two was defined as the fractal coefficient. Based on the Carman-Kozeny equation, the fractal equation of cross-section can be deduced, including the theoretical expressions on the fractal dimension of perimeter and area. A gas-liquid displacement model was established based on Washburn dynamic equations; the fractal equations of the interface position and velocity were obtained. The nano-micro pore structure of low-permeability coal was reconstructed based on Nano–CT and the pore size and volume distribution were obtained. The fractal dimension of the perimeter and area was calculated based on the experimental data. Finally, a two-phase of N2–H2 O displacement experiment was carried out based on NMR. Moreover, the distance and velocity distribution of the interface were obtained. The results show that the fractal structure of capillary model can be used as a geometric bridge to explain the mechanism of low permeability. The defined fractal coefficient can effectively reflect the influence of tortuosity and fractal dimension, as well as the linear relationship with permeability. The reliability of the fractal cross-section model was verified, and the low-permeability mechanism of coal should still consider the influence of fractal cross-section roughness. The validity of the fractal dynamic equation was confirmed by the distribution of water flooding signals by NMR.