A new improved depth-integrated flow equation for single rough-walled fractures, incorporating the roles of fracture mid-plane inclination

We present a theoretical development of an improved depth-integrated flow equation for single rough-walled fractures, which accounts for the role of fracture mid-plane inclination. We demonstrate that the role of fracture mid-plane inclination is more significant than the roles of divergence and convergence of fracture aperture as a source of inaccuracy in the Reynolds equation or lubrication approximation, a widely used approximation for Stokes flow in narrow domains. Simplified forms of the Stokes and continuity equations are locally depth-integrated along a direction normal to the fracture mid-plane in order to derive a local depth-integrated flow equation written with respect to a local inclined orthogonal surface coordinate system. A consistent transformation of the local depth-integrated flow equation into a global Cartesian surface coordinate system results in an improved depth-integrated flow equation written with respect to a global Cartesian surface coordinate system. Three examples of fracture domains, which have exact solutions for Stokes flow, are used to evaluate the accuracy of the improved depth-integrated flow equation and evaluate its improvement over the Reynolds equation. These examples unequivocally show that the improved governing equation leads to a significant improvement from the Reynolds equation in predicting the flow. The corresponding rigorous stochastic perturbation analyses for an effective transmissivity based on the improved flow equation demonstrate a practical utility of the improved flow equation for more realistic single rough-walled fracture domains. Preliminary Monte Carlo simulations based on the improved flow equation lead to values of the effective transmissivity in reasonable agreement with those predicted by the aforementioned stochastic analyses.