A general lumped parameter model for the interpretation of tracer data and transit time calculation in hydrologic systems

Abstract

We present a general lumped parameter mathematical model for hydrologic tracer data interpretation and mean transit time calculation in hydrologic systems. The model takes the form of the three-parameter gamma distribution and accounts for different mixing types: perfect mixing; no mixing (piston flow); partial mixing (dispersive mixing, or the type between perfect mixing and no mixing); and various combinations of the above types. In these combinations, the different mixing types simulated by the model conceptually represent reservoirs in series. We introduce the mixing efficiency to characterize the extent or degree of natural mixing in hydrologic systems. This parameter equals zero for piston flow (no mixing), unity for perfect mixing, and a value in between these two extremes for partial mixing. The general model simulates the combination of perfect mixing, partial mixing, and piston flow. Six other models that simulate one or two of these mixing types can be obtained as special cases from the general model. Therefore, seven models are introduced in this effort. Of these, four (including the general model) are new, and three are currently existing in the field of tracer hydrology. The three existing models are the perfect mixing model, piston flow model, and the perfect-piston flow model which simulates the combination of perfect mixing and piston flow. The new models are the perfect-partial-piston flow model (the general model), perfect-partial mixing model, partial-piston flow model, and partial mixing model.Modeled mean transit times for three case studies agree with previous estimates: 21 and 2.4 years for two springs (sites 2 and 45, respectively) on Cheju Island, Republic of Korea; and 3.0 years for the Ottawa River basin, Canada.