Analytical Lagrangian model of sediment oxygen demand and reaeration flux coevolution in streams

An analytical model is developed for unidirectional-flow waterways in which the dissolved oxygen (DO) mass balance is dominated by reaeration and sediment oxygen demand (SOD) fluxes. To accurately represent the feedback between the two principal fluxes and the resulting spatial distribution of depth-averaged DO concentration (CDO) in the water column, formulations for the fluxes are implemented that are consistent with mass transfer theory rather than commonly used formulations (e.g., zeroth-order SOD) that neglect mass transfer physics. Water-side and sediment-side processes are incorporated into the SOD formulation; the sediment-side processes are simplified and parameterized empirically. The resulting DO mass conservation equation is expressed as a first-order linear ordinary differential equation. The model has similarities to the classic Streeter-Phelps model in the following respects: (1) it implements a Lagrangian control volume, (2) it expresses the competition between two flux or source/sink terms in the DO mass balance, and (3) it applies downstream of a flow or DO introduction location. The analytical solution yields a steady-state longitudinal CDO profile that spatially evolves to an asymptotic condition whereby reaeration and SOD fluxes have equal values. The difference in CDO evolution when implementing a zeroth-order SOD formulation versus the first-order SOD formulation is highlighted. The flow management implications are discussed and an example calculation is presented for the case of flow augmentation in Bubbly Creek in Chicago, Illinois.