High-velocity flow in a rough fracture

Abstract

We simulate high-velocity flow in a self-affine channel with a constant perpendicular opening by solving numerically the Navier–Stokes equations, and analyse the resulting flow qualitatively and quantitatively. At low velocity, i.e. vanishing inertia, the effective permeability is dominated by the narrowest constrictions measured perpendicular to the local flow direction and the flow field tends to fill the channel due to the diffusion generated by the viscous term in the Stokes equation. At high velocity (strong inertia), the high-velocity zones of the flow field resemble a narrow tube of essentially constant thickness in the direction of flow, since the transversal diffusion is weak compared to the longitudinal convection. The thickness of the flow tube decreases with Reynolds number. This narrowing in combination with mass balance results in an average velocity in the flow tube which increases faster with Reynolds number than the average velocity in the fracture. In the low-velocity zones, recirculation zones appear and the pressure is almost constant.

The flow tube consists of straight sections. This is due to inertia. The local curvature of the main stream reflects the flow-tube/channel-wall interaction. A boundary layer is formed where the curvature is large. This boundary layer is highly dissipative and governs the large pressure loss (inertial resistance) of the medium. Quantitatively, vanishing, weak and strong inertial flow regimes can be described by the Darcy, weak inertia and Forchheimer flow equations, respectively. We observe a cross-over flow regime from the weak to strong inertia, which extends over a relatively large range of Reynolds numbers.