Exact analytical solution of the convolution integral for classical hydrogeological lumped-parameter models and typical input tracer functions in natural gradient systems

•A set of exact analytical solutions of the convolution integral is presented.•The synthetic functions are expressed in terms of elementary mathematical functions.•The synthetic functions are easily adapted to represent real input tracer functions.•Piston flow, Exponential, Piston-Exponential and Dispersion lumped models are used.•The solution only depend on the transit time and the assumed lumped model parameters.

SummaryThis work presents the analytical solution to the convolution integral by taking into account the most widely used lumped parameter hydrogeological models (Piston, Exponential, combined Exponential-Piston and Dispersion model) and the eight most typical input tracer functions (Constant; Sinusoidal with linear trend; Sinusoidal with combined sinusoidal and linear trend; Instantaneous pulse injection; Step or Heaviside; Instantaneous pulse with exponential ending; Long pulse with sharp ending; Long pulse with exponential ending) naturally occurring or usually conducted in aquifer systems under natural gradient conditions. For such cases, the output tracer function is expressed in terms of mathematical elementary functions that only depend on the aquifer mean transit time and the parameters belonging to the assumed lumped model.