This study provides a numerical approach to evaluate the effect of sample sizes on fluid flow through a single rough-walled fracture. The rough-walled fracture was generated by using mathematical statistics. Its aperture follows a normal distribution along a large-size plane of dimensions 20 m × 20 m, and from this large-size fracture plane, a series of fractures with different sizes were extracted. Then, a two-dimensional (2D) Reynolds equation was solved to simulate fluid flow through these rough-walled fractures with different mean fracture apertures and roughness. The impact of normal stress on the hydraulic properties was introduced by using a hyperbolic function. The simulated results show a positive correlation between hydraulic conductivity and fracture sizes. In fact, hydraulic conductivity of the fluid flow through a single rough-walled fracture increases with the increment of the fracture size until a given size from which it stays constant. The hydraulic conductivity for this given size can then be considered as a representative of that in larger sample sizes. This given size resulted to range from 12 m to 16 m, due to the variations in mean fracture apertures, roughness and normal stress. In addition, it was also found that this representative size tends to decrease with an increase in the mean fracture apertures, while tends to increase as the fracture roughness or the normal stress increases, since those variables affect the fracture topology and contact areas, which play an important role in controlling the tortuosity and connectivity of fluid flow through a rough fracture.

这项研究提供了一种数值方法，以评估样本量对通过单个粗糙壁裂缝的流体流动的影响。粗壁裂缝是通过使用数学统计产生的。其孔径沿尺寸为20 m×20 m的大尺寸平面遵循正态分布，并从该大尺寸裂缝平面中提取出一系列大小不同的裂缝。然后，求解二维（2D）雷诺方程，以模拟通过具有不同平均裂缝孔径和粗糙度的粗糙壁裂缝流过的流体。通过使用双曲函数介绍了法向应力对水力特性的影响。模拟结果表明，水力传导率和裂缝尺寸之间呈正相关。事实上，通过单个粗糙壁裂缝的流体的水力传导率会随着裂缝尺寸的增加而增加，直到给定尺寸为止保持恒定。对于给定大小的水力传导率，可以认为是较大样本量的水力传导率的代表。由于平均裂缝孔径，粗糙度和法向应力的变化，该给定尺寸的结果为12 m至16 m。此外，还发现该代表性尺寸倾向于随着平均裂缝孔径的增加而减小，而随着裂缝粗糙度或法向应力的增加而趋于增大，因为这些变量会影响裂缝的拓扑结构和接触面积，从而影响在控制通过粗糙裂缝的流体曲折度和连通性方面起着重要作用。