Abstract
We investigate the flow-wise variation of the hydraulic conductivity inside a non-uniformly shaped fracture with permeable walls. Using lubrication theory for viscous flows, in conjunction with the Beavers–Joseph–Saffman boundary condition at the permeable walls, we obtain an analytical expression for the velocity profile, conductivity, and wall permeation velocity. These predictions highlight the effects of geometric variation (through the local slope of the aperture’s flow-wise variation), the permeability of the walls (through a dimensionless slip coefficient), and the effect of flow inertia (through a Reynolds number). The theory is validated against an OpenFOAMⓇ solver for the Navier–Stokes equations subject to a tensorial slip boundary condition, showing good agreement. The mathematical results have implications on system-level (multiscale) modeling of hydraulically fractured reservoirs, in which the Darcy conductivity of each non-uniform passage must be accurately accounted for, throughout the fractured porous rock.
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