The hydraulic conductivity of shaped fractures with permeable walls

We derive the hydraulic conductivity $K$, i.e., the proportionally constant between the width-averaged velocity field and the pressure gradient in Darcy's law, for shaped fractures with permeable walls. As a model, we study a tapered Hele-Shaw cell, with a width gradient $dh/dx=\alpha$ in the flow direction, and porous boundaries. The permeable walls are treated using the Beavers--Joseph slip boundary condition. Using lubrication theory, we obtain $K$, accounting for geometric non-uniformity and leakage into the bounding surfaces. The approach is perturbative, giving both the leading-order term (independent of the Reynolds number $Re$) and the first correction in $Re$. Thus, our theory gives $K$ in terms of hydraulic parameters such as $Re$, geometric parameters such as the fracture's width $h(x)$ and $\alpha$, and the dimensionless slip coefficient $\phi$ at the porous walls. Previous research has not addressed the joint dependence on $Re$ and $\alpha$. Specifically, our calculations show that, quantitatively, $Re$ has a comparable effect to $\phi$ on the value of $K$, for $\alpha\ne0$. Finally, we use the open-source computational fluid dynamics software, OpenFOAM, to perform 3D direct numerical simulations to benchmark and verify our mathematical predictions.