Residence time, groundwater age, and solute output in steady-state groundwater systems

The total travel distance and the hydraulic conductivity in steady-state groundwater flow systems are modeled with asymmetric gamma distribution functions. The ratio of the total travel distance over the hydraulic conductivity, scaled by a factor involving the hydraulic gradient and the aquifer porosity, equals the residence time of groundwater. The residence time measures the period elapsed between recharge and discharge of groundwater from an aquifer flow system. The residence time is a random variable whose probability density function (p.d.f.) is determined by those of the total travel distance and the hydraulic conductivity. The p.d.f. of the residence time was derived and from that, applying basic mass-balance considerations, followed that of the groundwater age. The age is the time elapsed since the recharge of groundwater in transit through an aquifer flow system. It is shown that the p.d.f.s of the residence time and the groundwater age are related by a first-order, linear, differential equation. The cumulative distribution functions of the residence time and the groundwater age and their first and second moments are also derived, and so is the output concentration of a solute that enters a steady-state groundwater flow system dissolved in recharge. A computational example illustrates the advantage of treating the residence time and the groundwater age statistically and constraining them with Darcyʹs law and mass-balance principles. Solute output shows a complex structure with stable and radioactive solutes in deceptively simple steady-state groundwater flow regimes.