Modeling and inverting reactive stream tracers undergoing two-site sorption and decay in the hyporheic zone

Performing stream-tracer experiments is an accepted technique to assess transport characteristics of streams undergoing hyporheic exchange. Recently, combining conservative and reactive tracers, in which the latter presumably undergoes degradation exclusively within the hyporheic zone, has been suggested to study in-stream transport, hyporheic exchange, and the metabolic activity of the hyporheic zone. The combined quantitative analysis to adequately describe such tests, however, has been missing. In this paper, we present mathematical methods to jointly analyze breakthrough curves of a conservative tracer (fluorescein), a linearly degrading tracer (resazurin), and its daughter compound (resorufin), which are synchronously introduced into the stream as pulses. In-stream transport is described by the one-dimensional advection-dispersion equation, amended with a convolution term to account for transient storage within the hyporheic zone over a distribution of travel times, transformation of the reactive tracer in the hyporheic zone, and two-site sorption of the parent and daughter compounds therein. We use a shape-free approach of describing the hyporheic travel-time distribution, overcoming the difficulty of identifying the best functional parameterization for transient storage. We discuss how this model can be fitted to the breakthrough curves of all three compounds and demonstrate the method by an application to a tracer test in the third-order stream Goldersbach in Southern Germany. The entire river water passes once through the hyporheic zone over a travel distance of about 200 m with mean hyporheic residence times ranging between 16 and 23 min. We also observed a secondary peak in the transfer functions at about 1 h indicating a second hyporheic flow path. We could jointly fit the breakthrough curves of all compounds in three monitoring stations and evaluated the parameter uncertainty of the individual and joint fits by a method based on conditional realizations of the hyporheic travel-time distribution. The approach gives insight into in-stream transport, hyporheic exchange, metabolic activity, and river-bed sorption of the stream under investigation.