A “Coulomb friction” model of two-phase flow in a rough fracture

An “imperfect” Hele-Shaw cell (IHSC) with random variations of the aperture provides a useful analogue for a rough fracture. For flow of two immiscible fluids with a single interface between the phases in an IHSC tilted with respect to the horizontal plane, with pressure control at the inlet, there are, in general, multiple equilibrium interface profiles. This leads to hysteresis (history dependence) of the interface evolution and finite energy dissipation even in the limit of infinitely slow (quasistatic) driving, due to Haines jumps between the equilibria.

We use a recently developed spectral method that predicts the interface evolution and energy dissipation in such a system with high accuracy and computational efficiency. We show that, given the inlet pressure, the set of equilibrium interface configurations forms a band with rough boundaries. This constitutes a “sticky region”: an interface starting within it only undergoes minor deformations (maintaining its overall position without moving as a whole), whereas an interface starting outside it advances to the nearest boundary of the region. Drawing analogy between this behaviour and that of an object in a well with dry (Coulomb) friction, we hypothesise — and confirm numerically — that if the motion of the interface is reduced to a single variable, the mean height, then the evolution of this variable follows a simple law akin to a combination of viscous and dry friction. We then proceed to study systematically how the “dry friction” coefficient depends on the properties of the cell’s roughness, such as the aperture variance and the correlation length. Our results may serve as an input to an upscaled model of flow in fractures, replacing the full aperture field (typically unknown) with continuum roughness parameters.