In order to study the effect of fracture geometry on the nonlinear flow properties in aperture-based fractures, a fractal model based on the self-affinity is proposed to characterize the three-dimensional geometry of rough-walled fractures. By solving the N–S (Navier–Stokes) equation directly, the relationships between the Forchheimer-flow characteristics, fractal dimension, and standard deviation of the aperture have been obtained. The Forchheimer equation is validated to describe the nonlinear relationship between flow rate and pressure gradient. For lower flow rate, the influence of the fractal dimension almost can be ignored, but the linear coefficient increases and the hydraulic aperture decreases with increasing standard deviation of the aperture, respectively. For larger flow rate, the nonlinear coefficient increases with the growth of the standard deviation of the aperture and fractal dimension. Thus, an empirical relationship between the nonlinear coefficient, fractal dimension, and standard deviation of aperture is proposed. In addition, the critical Reynolds number decreases with the increase of the standard deviation of the aperture and the fractal dimension, and the numerical results are generally consistent with the experimental data.

中文翻译：

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基于自仿射孔径的裂缝中非线性流动特性的数值模拟

为研究裂缝几何形状对基于孔隙的裂缝非线性流动特性的影响，提出了一种基于自相似性的分形模型来表征粗糙壁裂缝的三维几何特征。通过直接求解 N-S（Navier-Stokes）方程，得到了 Forchheimer-flow 特性、分形维数和孔径标准偏差之间的关系。Forchheimer 方程经验证可用于描述流速和压力梯度之间的非线性关系。对于较低的流量，分形维数的影响几乎可以忽略不计，但线性系数增加，水力孔径分别随着孔径标准偏差的增加而减小。对于更大的流量，非线性系数随着孔径和分形维数标准差的增加而增加。因此，提出了非线性系数、分形维数和孔径标准偏差之间的经验关系。此外，临界雷诺数随着孔径标准差和分形维数的增加而减小，数值结果与实验数据基本一致。