A Lagrange transport approach for water quality modelling

Modelling the propagation of dissolved compounds is a demanding task as the used model has to be able to predict friction and dispersion effects. Looking at some popular commercial and non-commercial programs for urban drainage modelling shows certain limitations. Some of the programs allow an explicit calibration of the dispersion coefficient, others do not allow the explicit calibration of the dispersion effects but provide other possibilities, and still others are not able to address dispersion at all. Programs that allow the explicit calibration of the dispersion effect sometimes still face the problem that the identification of a proper dispersion coefficient can be difficult because numerical dispersion can compensate for the physical effect. In cases with low dispersion (e.g. sewer systems) the numerical dispersion alone can already be to high (Huisman et al., 1999). Simulators like SWMM, which do not provide the possibility to define a dispersion coefficient directly, demand for changing the model structure to reproduce the physical effect. As each conduit serves as completely stirred tank reactor one possibility to address dispersion effects is to vary the number and length of the conduits representing the real sewers and therefore changing the volume of each reactor.

To overcome these problems we have developed a Lagrange based transport model, which is a fundamentally modified version of the Lagrangian time-driven method (TDM) described by Rossman and Boulos (1996). The model uses a moving segments approach; dispersion is calculated between segments which follows each other. The approach itself introduces no numerical dispersion so that an explicit determination of diffusion as a model parameter becomes possible (with no influence of the model structure). As a consequence, the model structure can be defined depending on physical boundaries only. Furthermore, the model formulation allows an easy extension with reaction models by solving a set of ordinary differential equations (ODEs) for each segment in each time step. We use a matrix notation for the quality model (as widely used in dynamic treatment plant modelling) in combination with a mathematical expression parser and ODE solver resulting in a flexible and adjustable water quality model that can be defined by the modeler itself.

The model has been implemented in different simulation programs, amongst others in the SWMM5 computational engine. It has successfully been tested against different tracer experiments conducted in sewer systems and rivers. The functionality of the reaction model is demonstrated by implementing a number of accepted water quality models (sedimentation and resuspension, extended Streeter-Phelps, Lijklema, and IWAs RWQM No 1).