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Berryman-Milton upper-bound (Eq. 19) for the permeability k of a Boolean model of flat cylinders with height h = 1 and radius r = 102, 103, 104vs concentration of obstacles q, in log-log plot (black). Solid lines in green: r → ∞ expansion (Eq. 23). Solid lines in pink: dilute limit expansion q → 0 (Eq. 22).