Beyond Poiseuille flow: A transient energy-conserving model for flow through fractures of varying aperture

Abstract

Large-scale models of fluid flow through fractures are almost exclusively based on Poiseuille flow (the cubic law). When the fracture aperture is time and/or spatially varying, flow is transient, and/or flow rates are modest (Re≥1), Poiseuille flow predictions can deviate substantially from the full Navier–Stokes solution and generally do not satisfy conservation of energy. A new Reduced Dimension Fracture Flow (RDFF) model is derived which more accurately predicts transient fracture flow for incompressible fluids with modest Reynolds numbers. The RDFF model is derived from the two-dimensional Navier–Stokes equations and yields a two-field model (fluid flux and pressure) governed by conservation of mass and momentum. The RDFF model conserves energy and is shown to include both Poiseuille flow and Forchheimer flow as limiting cases. The performance of the RDFF model is compared to the existing Poiseuille flow model and the solution to the full Navier–Stokes equations in three benchmarks problems. The RDFF model demonstrates complex transient and inertial behaviours not previously captured and produces up to 400% improvements in error over Poiseuille flow in steady-state flow conditions for 1≤Re≤100. The RDFF model is demonstrated to be superior in application where the Reynold’s number is modest and when aperture varies in space and/or time.