Development of depth-averaged nonlinear flow model for fractures considering heterogeneity and significant inertial effects

Abstract

Fluid flow through fractures is central to many engineering applications and geophysical processes. Although the intertwined effects of aperture heterogeneity and inertial forces on fluid flow have been interrogated in last few decades, it remains challenging to accurately predict fracture flow field at high pressure gradient conditions when inertial effects are significant. To address this challenge, we rigorously derived a depth-averaged nonlinear flow model by introducing a quadratic velocity term into the linear flow model (i.e., Local Cubic Law that describes a linear relationship between pressure gradient and fluid flux) at the mechanistic level. The supporting laboratory experiments and direct numerical simulations through six rough-walled fractures with varying roughness further confirmed the robustness of the proposed model. We found that the predictive model outperforms existing alternatives with acceptable relative errors in mean velocity (<7%) considering a broad range of Reynolds number (0.01–167), suggesting its superior performance of handling pronounced inertial effects. Lastly, we parameterized the new model by establishing an empirical function that related parameter (ω) in the model to measurable fracture properties (i.e., aperture); this makes the predictive model viable and straightforward for predicting local velocity fields in pressurized engineered environments.