A nonlinear flow model for rough fractures with geometric heterogeneity based on improved Izbash's law

Nonlinear flow through rough fractures, driven by inertial effects, has been widely observed in field and laboratory experiments. However, existing models for capturing such nonlinear flow are still inadequate, particularly in the presence of significant inertial effects. This study introduces a nonlinear flow model for rough fractures that simultaneously considers geometric modifications and inertial effects. The modification of fracture geometries is based on empirical models with measurable parameters, and various combinations of these empirical models are tested to identify the most suitable model that aligns well with the nonlinear flow model. The improved Izbash's law is innovatively employed to incorporate inertial effects due to its similarity with the classic modified cubic law. Numerically generated fractures, featuring diverse levels of heterogeneity in the aperture field, are utilized to validate the performance of the proposed model. The mean flow velocity at the outlet of the proposed model is compared with that from empirical models and direct simulations using the Navier-Stokes equation (NSE). The Reynolds number (Re) ranges from 0.01 to 200, including significant inertial effects within these fractures. The ultimate nonlinear flow model achieves the lowest overall relative error for all of the fractures. The results show that the majority of the relative errors fall below 10% across this wide range of Re values, demonstrating the superior performance of the proposed model in capturing nonlinear flow in rough fractures. In contrast to the previous models, the proposed model can simulate nonlinear flow with significant inertial effects using simple geometric parameters. This feature potentially enables the incorporation of diverse flow regimes in more complex scenarios, such as in fracture networks.