The process of fluid flow in rock fractures involves complicated dynamical behavior of fluids, and its modeling is a challenge to numerical methods. In this paper, the Smoothed Particle Hydrodynamics (SPH), a particle method based on Lagrangian formulation, is employed to simulate the fluid flow in rock fractures by solving the Navier-Stokes equations directly. Firstly, the SPH method and the boundary treatment method used in this simulation were introduced and the computer code of SPH was developed and validated by a series of numerical benchmark tests with analytical solutions. Then simulations were carried out to investigate the fluid flow both in single fractures and intersected fractures, with and without considering effects of surface roughness. The results of the simulations are discussed and compared with the analytical solution by using the Cubic law derived from the Reynolds equation. The results show that in both of rough single fractures and fracture intersections, although the relationship between mean velocity and the Reynolds number is still linear, the solutions by using Cubic law overestimated the mean fluid velocity with increasing the Reynolds number, indicating possible underestimate of travel time of mass transport in the fracture network models.


Fractures are of great importance in geotechnical engineering, hydrogeological practice and environmental assessments because they provide pathways which usually take predominantly place for fluid flow in fracture rock systems. For better understanding the behavior of fluids flow in rock fracture systems, the basic understanding of fluids flow in single fracture and fracture intersections is needed.

In the research of fluid flow through rock fractures, numerical modeling is an important supplement to experimental investigation. The basic governing equations of general fluid dynamics are Navier-Stokes equations, but they are a set of nonlinear differential equations that is not easy to solve directly by traditional grid-based methods like Finite Element method (FEM) and Finite Volume Method (FVM), especially when it considering the real rock fractures in 3 dimensions and with complex geometry boundaries. So in the past researches, the Reynolds equation, a linearized simplification of Navier-Stokes equations, were often used in numerical studies (Zimmerman 1996). Although the Reynolds equation is much easier to solve, it may fail in accurate describing of the nonlinear flow phenomena (Yeo et al. 2005; Koyama et al. 2008; Tong et al. 2010).

In this research, the Smoothed Particle Hydrodynamics (SPH) was used to simulate the fluid flow in rock fractures by solving the Navier-Stokes equations directly. A boundary treatment method was proposed, and a SPH code was developed and tested by a series numerical benchmark tests with available analytical solutions with simplified model geometry and boundary conditions for fluid flow in simplified single fracture with roughness and fracture intersections. Differences between simulated results and analytical solutions based on Reynolds equations (Cubic law) are compared in this paper and the limitations of Cubic Law and Reynolds equation for flow in fracture network models are discussed.

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