Lagrangian modeling of water quality and contamination analysis of a St. Lawrence River reach : Lake St. Pierre (Canada)
Context The St-Lawrence Center, part of Environment Canada, undertook a few years ago the very ambitious project of studying the toxic contamination of the St-Lawrence River. In collaboration with the Institut National de la Recherche Scientifique - Eau, a sub-project based on numerical modeling was defined in order to analyze contaminant propagation from industrial and municipal effluents into the river system. Goals The specific goals of the project were the following : 1) to provide a precise quantification of contaminant concentrations in the effluent plume al a convenient scale; 2) to analyze areas influenced by main tributaries and different water masses entering the river reach; 3) to map and quantify areas as compared to water quality criteria ; 4) to provide a method to select relevant hydrological events as a significant part of the analysis framework Methodology Some basic choices were made at the beginning of the project : 1) the analysis framework emphasis the instream water quality instead of the effluent water quality; 2) numerical modelling was the main tool used to evaluate the water quality; 3) as far as possible references to public regulations were incorporated; 4) a strong complementarity of different computer tools was favoured : Geographical Information Systems, Database management systems, simulation models; 5) the numerical solution method for the transport diffusion model is typically Lagrangian : the Random Walk Method; 6) the contamination analysis uses the so-called « Weighted Unusable Area » method to quantify areas that do not respect some water quality criteria. A typical contamination analysis project based on numerical modelling includes the following steps (fig 2) : 1) a preliminary study to determine the main characteristics of the problem and to choose the best strategy to analyze it; 2) field measurements essential to the calibration and validation of the computer model; 3) hydrodynamic modelling provides the basic data on the flow field; this step includes the calibration and the validation of the model, as well as the prediction of the flow fields corresponding to well-defined and contamination relevant hydrological events; 4) hydrological analysis identifies the relevant flow events chat will further be used in the mode) prediction ; this approach allows standardization of this very important input data set and avoids arbitrary choices of flow field; 5) transport-diffusion modelling constitutes the main step; it provides the chemical species concentrations downstream from the effluent discharge and affords an estimate of the overall water quality of the reach, as influenced by the main tributaries. This step includes the calibration and the validation of the model which precedes the prediction exercise; 6) contamination analysis necessitates the choice of appropriate and relevant water quality criteria ; we propose a new approach, inspired by the Instream Flow Incremental Methodology often used to define the quality and availability of fish habitat in river reaches, to implement this step. Numerical methods As previously mentioned, the project included the development of a Lagrangian model to simulate the transport of solutes in a two-dimensional steady-state river flow. We will emphasize this point. The main objective of the software development was to provide an efficient and user-friendly management tool for the public agencies. Many analytical test cases helped in the choice of the best numerical algorithms, non-physical related parameters, and in the validation of the computer code. Furthermore, the results of two dye tracing experiments performed in conjunction with airborne remote sensing techniques provided data to validate the model on the St-Lawrence River (fig. 5, 6, land 8 illustrate different simulation results corresponding to the different tasks mentioned previously). In the next paragraphs, we will summerize the basic mathematical and numerical concepts implemented in the simulations. To simulate solute transport in water media (porous or free surface), one usually uses eulerian methods which lead directly to concentration values. The solution algorithm presented here is rather based on a Lagrangian method which offers an explicit control over the additional numerical diffusion associated with every discretization method. This approach, also called the Random Walk Method (illustrated in fig. 3), or Particle Tracking Method, is more and more often used to solve hyperbolic equations. So far, the literature does not provide many applications of this method to solute transport in free surface flow. Oil spin modeling is a domain where many applications have been reported. The propagation of solute matter in free surface flow is mathematically described with momentum, mass and solute conservation equations. Since the Random Walk solution method of the transport-diffusion equation (equ. 1) requires hydrodynamic data to calculate the mean transport on streamlines along with dispersion, independent simulations providing the necessary flow field data (velocities, diffusivities, depths) have to be performed before undertaking the transport-diffusion tasks. For this purpose, the Navier-Stokes shallow water equations have become a well known tool to represent flow field in shallow waters. However, one should be aware of some often neglected but important aspects of such models, such as moving boundaries and turbulence closure. Solution techniques Two main goals were kept in mind during the implementation of the various algorithms : precision of results and fast computation. The following choices were made to achieve these objectives : 1) A finite element discretization and solution method provides and carries hydrodynamic Information, but particles are tracked on a finite-difference grid (mixed discretization principle). 2) The convective component of the movement is realized by moving the grid instead of the particles (shifted grid principle). 3) Computation of concentrations optimizes smoothing while minimizing artificial diffusion (controlled effusive smoothing principle). 4) When a section of the plume is described in a steady state « regime », it is mot necessary to continue the simulation on that section to proceed downstream ; the simulation is divided in almost independent sections (convolution principle). 5) The particles have an a priori nondimensional weight and a unit concentration is calculated from these (unit plume principle). 6) The real concentration is linearly dependent on the pollutant loads introduced into the milieu (linearity principle). The Weighted Unusable Area Method The Weighted Unusable Area method provides a convenient means to compare effluent plume water quality to water quality criteria as well as to quantify areas that do not comply to them. A comparable method is widely used to define the quality and availability of fish habitat downstream from regulation reservoirs, with the purpose of establishing minimum guaranteed flow discharge to protect target species (the Instream Flow Incremental Methodology : IFIM). The method consists essentially of computing areas within the analysis domain weighted by a certain factor that represents the exceedence of certain water quality, criteria. Among different options to define the weighting factor, all incorporating the effective contaminant concentration, we defined the following : 1) the ratio of the concentration to the water quality criterion without consideration of exceedence or compliance; 2) weighting factor equal to 1 only if the concentration exceeds the criterion (non-compliance); 3) option #1, but using the concentration results corresponding only to the effluent plumes excluding the ambient water quality of the reach ; this emphasizes individual corporate responsibility (proposed for implementation); 4) option 11, but with the ratio increased by a power « n », a procedure that emphasizes the non-linear increase of toxicity related to the exceedence of the criterion (could be useful for academic purposes). We also propose a Global Weighted Unusable Area concept to combine all the different chemical species present in an effluent plume. The combination is made possible using the specific criterion corresponding to each species. This procedure leads to a new state variable that represents Contamination Standard Units.