Effective permeability of complementary wedges ($k_{\mathrm{eff}}=(k_1-k_2)/\log (k_1/k_2)$)

In a slender layer, two parallel sublayers of fractions $x/L$ and $1-x/L$ have local effective permeability $K(x)=k_{1}x/L+k_2(1-x/L)$. Integrating their successive hydraulic resistance gives the logarithmic mean displayed above. A leading pressure gradient uniform across the thickness does not make parcel paths horizontal; a small transverse velocity preserves mass conservation as the cross-sectional conductance changes.