A geometric parameter linking aperture heterogeneity and drainage morphology in rough fractures

The immiscible displacement of a wetting fluid by a non-wetting fluid (drainage) in rough-walled fractures is central to subsurface applications such as CO₂ sequestration and underground hydrogen-storage. Predicting displacement outcomes, including residual trapping and invasion morphology, requires understanding how viscous and capillary forces interact with fracture-scale geometric heterogeneity. In horizontal systems, where gravity effects are negligible, this interaction gives rise to complex and spatially variable invasion patterns. In porous media, drainage morphologies are commonly interpreted using the viscosity ratio M and capillary number Ca within the classical framework of [1]. However, even within this framework, the influence of structural heterogeneity on displacement patterns remains unresolved. In rough fractures, aperture variations introduce an additional geometric control that further complicates this picture. The competition between smoothing by in-plane interfacial curvature and roughening induced by out-of-plane aperture variability makes invasion morphologies difficult to predict [2]. As a result, the commonly used roughness measure, the closure δ, is insufficient: fractures with identical δ can exhibit markedly different invasion patterns under identical flow conditions.A more complete geometric description must therefore account for both the amplitude and spatial organization of aperture variability. The fracture aperture field a(x,y), is characterized not only by its variance σa but also by a lateral correlation length Ic relative to the fracture length L. Focusing on the out-of-plane contribution to capillary pressure, which scales with ,1/a(x,y) we derive a dimensionless geometric parameter quantifying capillary heterogeneity. Introducing Ic as the characteristic lateral scale of variability leads to a dimensionless parameter which links aperture variance and spatial correlation to capillary pressure fluctuations. This formulation revisits the curvature-ratio introduced by [2], while reformulating it in terms of statistically measurable aperture variability, yielding a practical geometric measure of capillary heterogeneity.Direct numerical simulations of horizontal drainage were performed using a validated Volume-of-Fluid framework [3] for Ca between 10-2 and 10-5 and M=0.1, 0.8. Synthetic self-affine fractures with systematically varied mean aperture, aperture variance, and correlation length were considered. Displacement morphology was quantified using fractal dimension, fluid-fluid interfacial length, and typical finger width. Preliminary results show that fractures sharing similar values of the aforementioned dimensionless parameter, exhibit comparable invasion structures, regardless of how the roughness is generated, indicating that this parameter provides a physically grounded link between fracture geometry and drainage morphology.[1] Lenormand, R., Touboul, E., & Zarcone, C. (1988). Numerical models and experiments on immiscible displacements in porous media. Journal of fluid mechanics, 189, 165-187.[2] Glass, R. J., Rajaram, H., & Detwiler, R. L. (2003). Immiscible displacements in rough-walled fractures: Competition between roughening by random aperture variations and smoothing by in-plane curvature. Physical Review E, 68(6), 061110.[3] Krishna, R., Méheust, Y., & Neuweiler, I. (2024). Direct numerical simulations of immiscible two-phase flow in rough fractures: Impact of wetting film resolution. Physics of Fluids, 36(7).