Perturbation Solutions for Flow in a Slowly Varying Fracture and the Estimation of Its Transmissivity

Abstract

Flow in fractures or channels is of interest in many environmental and geotechnical applications. Most previously published perturbation analyses for fracture flow assume that the ratio of the flow in the fracture aperture direction to the flow in the fracture length direction is of the same order as the ratio of mean fracture aperture to fracture length, and hence, the dominant flow is in the fracture length direction. This assumption may impose an overly strict requirement for the flow in the fracture length direction to be dominant, which limits the applicability of the solutions. The present study uses the ratio of aperture variation to length as the perturbation parameter to derive perturbation solutions for flow in two-dimensional fractures under both the pressure boundary condition (PBC) and the flow rate boundary condition (FBC). The solutions are cross-validated with direct numerical solutions of the Navier–Stokes equations and with solutions from published perturbation analyses using the geometry of two-dimensional symmetric wedges and fractures with sinusoidally varying walls. The study shows that compared with the PBC solution, the FBC solution is in a closer agreement with simulation results and provides a better estimate of the fracture transmissivity especially when the inertial effects are more than moderate. The improvement is due mainly to the FBC solution providing a more accurate quantification of the inertial effects. The solutions developed in this study provide improved means of analysing the hydraulic properties of fractures/channels and can be applied to complex flow conditions and fracture geometries.