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

Abstract

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. IntroductionThis research deals with the development from first principles of a dynamical model for concentration-discharge (C-Q) dynamics in runoff, subject to dual time-varying inputs. Although multiple reservoir models are widely used, they are often formulated in an ad hoc fashion, particularly as regards the number of required reservoirs (i.e., state variables) and the form of constitutivc relations. The result is that the relation of such models to the underlying physics is obscure. The dynamical systems perspective taken here, applied in conjunction with appropriate continuum theories, provides a consistent conceptual framework in which to both derive and test the validity of such models. A comprehensive approach would formulate a dynamical theory for modeling the fully coupled hydrochemical response of saturated and unsaturated storage discharging to streams, rivers, and lakes for distributed inputs, including chemical transformations. The present paper extends the work of Gelhat and Wilson [1974]

Bridging river basin scales and processes to assess human‐climate impacts and the terrestrial hydrologic system

“…Most hydrologic models only represent the pressure gradients within water resources systems and do not attempt to represent flow paths or residence times explicitly [Brandes et al, 1998;Duffy and Cusumano, 1998;Weiler et al, 2003]. Reproducing the main system modes with respect to the pressure response can often be achieved with very parsimonious models [Young, 2001], but new modeling approaches are required to advance our understanding of surface-atmosphere-groundwater interactions as well as solute transport.…”

Inferring the effect of catchment complexity on mesoscale hydrologic response

“…This ''hysteresis'' [Haines, 1930;O'Kane and Flynn, 2007] or ''lag effect'' has been attributed to (1) the flushing of solutes at the beginning of storm events [Walling and Foster, 1975], (2) the different arrival times of runoff contributions originating from hydrochemically contrasting areas or subbasins [Walling and Webb, 1980] and (3) the kinematic wave effect [Lighthill and Whitham, 1955;Kurtenbach et al, 2006], where the flood wave propagates faster within the river network than the water body and the associated solutes. Solute transport time scales are important controls on looping orientation and direction of C-Q relationships [Duffy and Cusumano, 1998;Hornberger et al, 2001]. Rice et al [2004] use a discriminant function analysis to characterize C-Q loops and to describe environmental conditions as responsible controls.…”

Effects of Dynamic Forcing on Hillslope Water Balance Models

“…Duffy and colleagues studied equilibrium solutions of a continuum model of unsaturated flow in a simplified hillslope [15,16,6] in order to extract a simplified model of a single hillslope for both flow and species transport. Numerical solutions of continuum equations on hillslopes have also been used to study the dependence of saturated area formation on topographic factors, soil properties, and rainfall intensity [28].…”

“…This formal framework for finite dimensional models of watersheds could conceivably unite many of the useful features of process models into a more rigorous physical and mathematical theory based on volume and time averaging as well as thermodynamically constrained closure of the balance equations. Our approach will take the time-averaging approach used in the framework and examples presented in [31,33,32] to derive models quite similar to the water balance models in [15,16,6]. …”