A method of calculating pumping induced leakage

We have discussed pumping induced leakage in a leaky-confined aquifer bounded by a relatively thin aquitard whose storage effect is neglected. Both constant-rate and constant-drawdown, fully penetrating vertical pumping wells are considered. We have derived closed-form analytical solutions of the steady-state leakage rates and volumes within any given radial distance from the well for both the constant-rate and constant-drawdown pumping wells. For the constant-rate well, we have derived the closed-form analytical solutions for the total leakage rate and volume over the entire aquitard–aquifer boundary at a given time; and the semi-analytical solutions for calculating the leakage rate and volume within any radial distance from the pumping well at a given time. For the constant-drawdown well, we have further provided the analytical solutions of the leakage rate and volume in the Laplace domain; those solutions are numerically inverted to yield results in the real time domain. With the scale-invariant relationship of [Butler Jr., J.J., Tsou, M.-S., 2003. Pumping-induced leakage in a bounded aquifer: an example of a scale-invariant phenomenon. Water Resour. Res. 39(12), 1344, doi: 10.1029/2002WR001484.], the solutions of the total leakage rates and volumes can be generalized to finite size aquifers with lateral impermeable boundaries. These solutions are useful for managing multi-layer aquifers in which pumping induced water transfer across the aquitards can occur.