A Simple Analytical Formula for the Leakage Flux Through a Perforated Aquitard

In this methods note, we present a simple analytical formula to quantify the steady‐state leakage flux (Q) over a perforated aquitard. The flux depends on the aquitard thickness (D), the radius of the perforation (R), the hydraulic conductivity of the material inside the perforation (k_fill), the conductivities of the overlying and underlying aquifers (k₁ and k₂, respectively), and the head difference between the two aquifers (ΔH): This equation assumes an aquitard separating two homogeneous and infinite aquifers (R ? aquifer thickness) in which radial flow to and from the perforation occurs, with no other recharge or discharge boundaries near the perforation. The flux through a perforation in a hypothetical case study with D = 10 m, k₁ = 10 m/d, k₂ = 20 m/d, R = 0.072 m, and ΔH = 1 m ranges between less than 1 mL/d if the hole is backfilled with bentonite (k_fill = 10^?4 m/d), to several liters per day if the perforation is backfilled with sand from the overlying aquifer (k_fill = 10 m/d), to several m³/d if the perforation forms an open conduit (k_fill = 10⁵ m/d). The leakage fluxes calculated with this model agree well with those calculated using a numerical model (MODFLOW).