Mathematical model development for non-point source in-stream pollutant transport

Non-point source pollution is a primary cause for concern globally. Various models have been developed to tackle this situation with much emphasis placed on best management practices. This practice has, however, proven to be insuffi cient to solve the NPS pollution situation. Existing non-point source models are watershed-based and complicated both in operation, parameter estimation and data requirements. A non-point source model is proposed using the concept of the Hybrid Cells in Series model. The model is a three-parameter model made up of three zones, which describes pure advection through time delay in a plug zone, with combined advection and dispersion occurring when the other two zones are considered as thoroughly mixed. The proposed model is tested using synthetic data and fi eld data from the Snake River, Colorado, USA, obtained from literature. Simulations were performed at four sample points; two from the tracer injection point along the Snake River before a confl uence and two further downstream after the confl uence. A regression analysis was carried out to determine the model’s capability to simulate pollutant transport for the four sampling points. The coeffi cients of determination are 0.98, 0.94, 0.84 and 0.97 while the standard error for each reach is 2.28E-2, 2.70E-2, 2.32E-2 and 9.35E-3 respectively. The results show good agreement between the measured and the simulated data. The response of the C-t profi les produced by the proposed model for both synthetic and fi eld data demonstrates its ability to eff ectively simulate pollutant transport in natural rivers subject to non-point source pollution. 92 J. Adu, M.V. Kumarasamy Germany. These models are predominantly used for evaluation of NPS pollution with relation to rainfall-runoff processes, soil erosion, sediment transport, pollutant source identifi cation and migration within catchments (Chen et al. 2013, Ouyang et al. 2016). The investigation into behaviour patterns of NPS pollutant clouds within water bodies is uncommon. Study on the hydrodynamic processes which occur within a water body due to NPS loading, and the rate NPS pollutant concentration patterns distribute in-stream cannot be ignored. The diffi culties associated with eff ective prediction of in-stream fl ux of NPS pollutants is that requires a simplifi ed and easy to interpret model, which must be able to balance the model complexity with available data. Theoretical Framework Mathematical models are generally based on the fundamentals of conservation of mass which combined with Fick’s law of diff usion represents the spatial and temporal eff ects of advection and dispersion of contaminants in water bodies. Research over time identifi ed limitations in the foremost Advection-Dispersion Equation (ADE) leading to the development of other models like the Cells in Series (CIS) and Aggregated Dead Zone (ADZ) models. The practical application of ADE is fl awed due to its inability to adequately interpret observed tracer profi les where uniform mixing due to turbulence occurs (Lees et al. 2000). Accuracy in simulating tracer profi les which are dependent on the dispersion coeffi cient within the stream has also been observed as a fl aw of the ADE model (Bencala and Walters 1983, Lees et al. 2000, Ghosh et al. 2004, 2008, Muthukrishnavellaisamy et al. 2009). The CIS model was considered as an alternative to the ADE which reduces the second-order diff erential equation of the ADE to a fi rst-order diff erential equation. However, as the CIS model considers individual cells as thoroughly mixed, the advection component is insuffi ciently simulated, thus, limiting its use in natural streams (Ghosh et al. 2008, Kumarasamy 2015). The ADZ model recognizes storage zones in stream beds and banks, which is considered a signifi cant factor for dispersion within water bodies. It decouples advection and dispersion which allows pure time delay for advection to occur. Thus, adequately describes observed concentration profi les in natural streams. However, parameter estimation and determination of the model orders are the primary challenges with ADZ (Lees et al. 2000; Ghosh et al. 2008). Other models which have been considered as suitable alternatives to ADE model, include the Transient storage model (TS) (Bencala and Walters 1983) modeled in line with the ADZ model, and the Hybrid Cells in Series model (HCIS) (Ghosh et al. 2004, 2008). Transient Storage (TS) Model The transient storage (TS) model is a modifi cation of the aggregated dead zone (ADZ) model and adds the process of transient storage to ADZ, which accounts for the elevated tails and skewness in its C-t profi les. It considers shear fl ow in uniform channels to simulate dispersion within a stream. TS model divides fl ows in rivers into two zones comprising the stream channel where transport mechanisms of advection and dispersion occur, and a transient storage zone where portions of solutes are trapped and isolated from the stream channel in underfl ow channels. It is conceptualized on the basis that as the pulse of solute moves along a stream, the concentration of the solute in the stream channel is attenuated while some solute mass remains within the storage zones. As the pulse passes through the stream channel, the retained solute in the storage zone is released back into the stream resulting in a gradual tailing of the C-t profi le. The governing equations of TS model describing a one-dimensional solute transport under steady and uniform fl ow conditions are: s C C C u E C C t x x (1)