A low‐dimensional model for concentration‐discharge dynamics in groundwater stream systems

Abstract

In this paper we investigate the physical basis and validity of a dynamical model for environmental tracer response for a groundwater‐dominated stream reach or small catchments. The dynamical model is formed by volume averaging of the local equations of saturated flow and solute transport. The approach interprets the empirical concentration‐discharge C‐Q as a pair of integrated state variables from an underlying state space (or phase space) for the hillslope or catchment response. The inputs represent episodic, seasonal, and random recharge rate and concentration time series. A closed form solution is found for constant inputs. Results are compared with numerical solutions of the governing partial differential equations, and agreement is found for a full range of initial states and levels of forcing. For pulse‐type or piecewise constant recharge events, the phase plane trajectories for C‐Q are shown to exhibit looping behavior, without the need for an assumption of hysteresis in the model. The orientation and looping direction of these solutions are shown to be controlled by the dimensionless ratio of solute residence time to hydraulic relaxation time and the relative phase lag between recharge and recharge concentration. The closed form solution is extended to the case of piecewise constant, random, input sequences resulting in a “random map” for C‐Q. An application is presented for seasonal storage and flushing of SO−4 in runoff for a small catchment in central Pennsylvania.