Lubrication theory analysis of the permeability of rough-walled fractures

Lubrication theory is used to study the permeability of rough-walled rock fractures. In this approximation, which is valid for low Reynolds numbers and under certain restrictions on the magnitude of the roughness, the Navier-Stokes equations that govern fluid flow are reduced to the more tractable Reynolds equation. An idealized model of a fracture, in which the roughness follows a sinusoidal variation, is studied in detail. This fracture is considered to consist of a random mixture of elements in which the fluid flows either parallel or transverse to the sinusoidal bumps. The overall permeability is then found by a suitable averaging procedure. The results are similar to those found by other researchers from numerical analysis of the Reynolds equation, in that the ratio of the hydraulic aperture to the mean aperture correlates well with the ratio of the mean aperture to the standard deviation of the aperture. Higher-order approximations to the Navier-Stokes equations for flow between sinusoidal walls are then studied, and it is concluded that in order for the lubrication approximation to be valid, the fracture walls must be smooth over lengths on the order of one standard deviation of the aperture, which is much less restrictive a condition than had previously been thought to apply.