Dilute permeability enhancement by aligned cracks ($k_{xx}^{\ast }=k\left[1+\pi \varphi a^2\frac{\alpha }{1+\alpha }+O((\varphi a^2{)}^2)\right]$)

At small areal number density $\varphi$, identical aligned cracks contribute independently the fixed-throughflow dissipation reduction of a conductive crack. Equating the mean dissipation density to $\mu U^2/k_{xx}^{\ast }$ gives the displayed first-order enhancement. Writing $k/[1-\pi \varphi a^2\alpha /(1+\alpha )]$ is an algebraic noninteraction estimate, not a claim of accuracy at higher density. The enhancement saturates as crack conductance becomes large compared with transport through the surrounding porous matrix.