Abstract
Previous studies pointed out that the hydraulic aperture (b
h) is solely dependent on the geometric features of a fracture, independent of fluid inertia effects. Here we present an inertial hydraulic aperture (b
ih) that considers the fluid inertial effect and fracture geometry effect by massive direct numerical simulations of fluid flow in real and artificial 3-D fractures. Simulation results indicate that with an increase in Reynolds number (Re), the evolution eddy volume ratio exhibits three distinct stages: stable stage (Re < 1), fluctuating stage (1 ≤ Re ≤ 10), and increasing to stable stage (Re > 10). These stages correspond to the transition of flow regimes from the viscous Darcy regime to the weak inertia regime, and further developing into the strong inertia regime. Among them, Re = 1 can be considered as the critical point for the onset of the non-Darcy flow. Furthermore, As Re increases, the evolution of b
ih exhibits four stages influenced by fluid inertia effects and main flow width in the fracture: stability, slight increase, slight decrease, and rapid increase. Then, based on 892 sets of simulation results (Re ≥ 1), the expression of b
ih was obtained using Gene Expression Programming. Compared to the four existing empirical models of b
h, the present b
ih exhibits the highest accuracy and the lowest errors (R
2 = 0.994, MAE = 0.008, RMSE = 0.013). Finally, the proposed b
ih is further employed to modify the Forchheimer equation. This study enhances the understanding of hydraulic conductivity in 3-D rough single fractures.
