A numerical study on differences in using Navier–Stokes and Reynolds equations for modeling the fluid flow and particle transport in single rock fractures with shear

摘要

The study on fluid flow and transport processes of rock fractures in most practical applications involves two fundamental issues: the validity of Reynolds equation for fluid flow (as most often assumed) and the effects of shear displacements on the magnitudes and anisotropy of the fluid flow velocity field. The reason for such concerns is that the impact of the surface roughness of rock fractures is still an unresolved challenging issue. The later has been systematically investigated with results showing that shear displacement plays a dominant role on evolutions of fluid velocity fields, for both magnitudes and anisotropy, but the former has not received examinations in details due to the numerical complexities involving solution of the Navier–Stokes (NS) equations and the representations of fracture geometry during shear. The objective of this paper aims to solve this problem through a FEM modeling effort. Applying the COMSOL Multiphysics code (FEM) and assuming a 2D problem, we consider the coupled hydromechanical effect of fracture geometry change due to shear on fluid flow (velocity patterns) and particle transport (streamline/velocity dispersion), using measured topographical data of natural rock fracture surfaces. The fluid flow in the vertical 2D cross-sections of single rock fractures was simulated by solving both the Navier–Stokes and the Reynolds equation, and the particle transport was predicted by the streamline particle tracking method with calculated flow velocity fields (vectors) from the flow simulations, obtaining results such as flow velocity profiles, total flow rates, particle travel time, breakthrough curves and the Péclet number, Pe, respectively. The results obtained using NS and Reynolds equations were compared to illustrate the degree of the validity of the Reynolds equation for general applications in practice since the later is mush more computationally efficient for solving large-scale problems. The flow simulation results show that both the total flow rate and the flow velocity fields in a rough rock fracture predicted by the NS equation were quite different from those predicted by the Reynolds equation. The results show that a roughly 5–10% overestimation on the flow rate is produced when the Reynolds equation is used, and the ideal parabolic velocity profiles defined by the local cubic law, when Reynolds equation is used, is no longer valid, especially when the roughness feature of the fracture surfaces changes with shear. These deviations of flow rate and flow velocity profiles across the fracture aperture have a significant impact on the particle transport behavior and the associated properties, such as the travel time and Péclet number. The deviations increase with increasing flow velocity and become more significant when fracture aperture geometry changes with shear.