Semi-analytical solutions for transient flow to a partially penetrated well with variable discharge in a general three-layer aquifer system

•General semi-analytical model for flow in a three-layer aquifer system is developed.•The well of variable discharge is partially penetrating in any part of the middle layer.•Dimensionless drawdown solutions for various top and bottom boundaries are obtained.•Obtained solutions unify existing solutions for two-layer aquifer and confined aquifer.•Drawdown behaviors due to an exponentially decreasing discharge are explored.