There are many applications which require a fracture flow model that accounts for variations in aperture beyond surface roughness. Large-scale models and simulations of fluid flow through fractures are almost exclusively based on the cubic law (Poiseuille flow) for steady-state flow through rigid parallel plates. When the fracture aperture is time and/or spatially varying, flow is transient, and/or flow rates are modest (Re≥1), cubic law predictions can deviate substantially from true fluid behaviour. In this paper, we present a new Reduced Dimension Fracture Flow (RDFF) model which more accurately predicts transient flow for incompressible fluids with modest Reynolds numbers through fractures with time and/or spatially varying aperture. The RDFF model is derived from the two-dimensional Navier-Stokes equations and yields a two-field model (fluid flux and pressure) governed by the conservation of mass and momentum. The RDFF model is shown to conserve energy in spatially varying fractures where the cubic law does not. We demonstrate that the RDFF model captures complex transient and inertial behaviours not previously captured for flows with modest Reynolds numbers (1≤Re≤100) and demonstrates up to 400% improvements in error over the cubic law in steady-state flow conditions through fractures with sinusoidally varying aperture.
Large-scale simulations of flow through fractures rely almost exclusively on the Poiseuille flow model, also known as "the cubic law". The cubic law is the analytical solution to the Navier-Stokes equations for steady flow of an incompressible fluid through parallel plates. It provides a constitutive equation linking the pressure gradient to the fluid flux according to the relationship
(Equation)
in which p is the pressure, μ is the kinematic fluid viscosity, w is the fracture aperture, and q is the fluid flux. In hydro-mechanical models with deformable fractures, such as hydraulic fracturing or enhanced geothermal energy, the cubic law is often combined with the conservation of mass derived from a fracture-scale control volume:
(Equation)
in which q is the flux from the cubic law. Despite the assumption that the cubic law applies for rigid parallel plate geometries, the conservation of mass allows both time and spatially varying apertures.
