Gravity flow in rock fractures with substrate and edge drainage

We investigate the influence of fluid rheology on flow in a finite rock fracture with vertically varying aperture and subject to competing drainage mechanisms due to a permeable substrate and a draining edge. The flow is due to the release of a finite volume of fluid, and the rheology of the fluid is either Newtonian, Ostwald-deWaele, or Herschel-Bulkley. The Hele-Shaw analogy between lubrication and seepage flows allows extending our results to a porous medium with permeability and porosity varying in the vertical direction. The general solution is numerical, except for a self-similar solution derived for Newtonian fluids in a constant aperture fracture and another for Ostwald-deWaele fluids without substrate drainage. Results for the profile of the current and the volume remaining within the fracture, or drained at the substrate and edge, depend on a dimensionless parameter lambda incorporating fluid rheology, fracture geometry, and ambient depth; drainage times exhibit order of magnitude variations depending on lambda. A second dimensionless parameter, lambda', intervenes for Herschel-Bulkley fluids, with lambda' -> infinity for Ostwald-deWaele fluids. The theoretical model is validated with a series of experiments conducted with a novel experimental apparatus, accurately reproducing the condition of substrate drainage and allowing the experimental determination of lambda and lambda('). The agreement between theory and experimental results for both configurations with constant and V-shaped aperture is quite good, considering model approximations and experimental uncertainties. The present analysis shows how domain anisotropy, though simply schematized, and fluid rheology are relevant for the correct estimation of all integral variables, such as the residual fluid volume in the fracture as a function of time.