Bingham fluid flow in spatially variable fractures

Non-Newtonian fluid flow in fractured media is of interest to hydrologists, geophysicists, and mining engineers. Since laboratory and field investigations evidence a strong degree of variability in fracture aperture, a large body of literature is specifically concerned with evaluation of an equivalent aperture (or permeability), adopting different constitutive equations and aperture variability models. The equivalent aperture for non-Newtonian fluid flow is defined as the parallel plate aperture which would permit a given volumetric flux under an assigned pressure gradient, thereby generalizing the concept of hydraulic aperture used for Newtonian flow. In this paper, the Bingham model with yield stress t0 has been adopted to describe the fluid rheology; the aperture is taken to vary as a spatially homogeneous and correlated random field with a lognormal aperture density distribution of assigned mean <b> and variance s2. The equivalent fracture aperture is derived for a specific geometry where the flow is perpendicular to the aperture variation. Under ergodicity, results are obtained by discretizing the fracture into elements of equal aperture and assuming that the resistances due to each aperture element are in parallel. The equivalent fracture aperture is greater than the mean, and their ratio is found to depend on aperture variability, represented by log-aperture variance s2, and on a dimensionless parameter w, equal to the wall shear stress in a fracture with aperture equal to <b> divided by the Bingham yield stress. The ratio is weakly dependent on w, and tends to increase as s2 increases. When w tends to infinity, all our expressions reduce to those derived in the past for Newtonian flow and lognormal aperture distribution.