Beyond the cubic law: A finite volume method for convective and transient fracture flow

Abstract

AbstractThe reduced dimension fracture flow model (referred to as the GG22 model) is a recently derived extension of the cubic law model for flow through fractures of variable aperture with fluid inertia effects. Novel numerical methods are required to solve the nonlinear partial differential equations governing the GG22 model, as it is more complex than the cubic law. The GG22 model is derived from Navier–Stokes, which allows the adoption of similar numerical methods to the Navier–Stokes equations, but the model contains its own idiosyncrasies which must be addressed. This article presents the first numerical methods to solve the GG22 model. An explicit multi‐step finite volume method is developed and verified. The method is based on deriving a Poisson equation for pressure with an additional continuity correction to overcome numerical instabilities. The critical timestep is derived and shown to be a function of the fundamental frequency of the fracture‐fluid system and the maximum fluid velocity. The results show excellent agreement with analytical solutions, and the method demonstrates a first‐order rate of fluid flux convergence in time and a second‐order rate of pressure convergence in space. The model is applied to a traveling aperture wave which shows that higher pressures are required to generate lower average fluxes than predicted by the cubic law.