Inapplicability of some cubic law corrections for slow flows in channels with rough walls

The cubic law defines the relationship between the pressure drop, the average fluid velocity in a channel, and its width. This law, which assumes smooth walls, requires a correction for surface roughness to be applicable to real fractures with highly rough surfaces. We assessed the applicability of popular formulas and statistical parameters for calculating the hydraulic width using three-dimensional (3D) numerical modeling of flow in channels with periodic wall roughness. The results show that the dependence of the hydraulic width on the characteristic roughness length generally cannot be adequately captured by a single statistical parameter. Calculating local width and its statistics captures this dependence but introduces errors in absolute values. The root mean square deviation of the surface slope effectively describes the hydraulic width for parallel-walled channels but fails for other shapes, requiring additional parameters. Most importantly, we demonstrate that analytical formulas based on two-dimensional (2D) flow equations or lubrication equations are not always valid, even when the flow appears to be two-dimensional.