Hydrodynamic Studies and Dynamic Parameter Estimation for Eutrophication Models
Sustainable process-product development & green chemistry
Environmental Engineering & Management (T1-3)
Keywords: eutrophication, mathematical model, phytoplankton, parameter estimation
The increasing inflows of nutrients into lakes and reservoirs, mainly due to agricultural activities, have intensified eutrophication of water bodies, which has in turn paved the way to the development of detailed ecological water quality models. These models provide a representation of major physical, chemical and biological processes that affect the biomass of phytoplankton and nutrients throughout periods of time and, once calibrated, they allow the prediction of the system evolution and consequent actions for remediation can be taken.
Eutrophication models comprise a set of complex nonlinear partial differential algebraic equations, with rate coefficients that require calibration to suit site-specific conditions. Consequently, the first step in an eutrophication model development is the resolution of a parameter estimation problem. Zhang et al. (2004) have proposed a sequential procedure to determine phytoplankton and zooplankton parameters using exergy as the objective function and calibrating both physical and chemical parameters by trial and error. More recently, Shen (2006) proposed a least-squares objective function and the resolution of the dynamic parameter estimation problem through the application of a modified Gauss-Newton method capable of handling upper and lower bounds on parameters and the Hessian being approximated with information from the sensitivity matrix calculated by finite differences.
In this work, we have formulated a parameter estimation problem with a least-squares objective function subject to a partial differential algebraic equations (PDE) model resulting from temporal and spatial dynamic mass balances in the phytoplankton groups diatoms, green algae and cyanobacteria; dissolved oxygen and nutrients, such as nitrate, ammonium, organic nitrogen, silica, phosphate and organic phosphorus. Algebraic equations represent profiles for temperature, solar radiation and river inflows, in addition to the calculation of most factors that affect rate equations, such as effect of solar radiation, nutrients, etc. The PDE is transformed into an ordinary differential equation system by applying the Method of Lines to spatially discretize the PDE into sets of ordinary differential-algebraic equations (DAE) (Rodriguez and Diaz, 2006). The DAE optimization problem is then transformed into a large nonlinear programming (NLP) problem by representing state and control variables profiles by polynomial functions over finite elements in time. The model has been formulated within the GAMS modeling environment and the NLP problem has been solved with a successive quadratic programming algorithm.
Hydrodynamic information required for the ecological model was obtained running the academic program DYSREM (DYnamic REServoir Simulation Model) (Antenucci, 2000), a one-dimensional hydrodynamics model for predicting the distribution of temperature along the column height, salinity and density, satisfying the one-dimensional approximation.
Field data with a twice a week frequency throughout a year have been obtained by the authors (Parodi et al., 2005) in Paso de las Piedras Reservoir (38° 22´ S and 61° 12´ W), which supplies drinking water to more than 400,000 inhabitants in Bahia Blanca, Argentina. At present, the trophic level of this lake corresponds to eutrophic category and it undergoes repeated blooms of algae (Parodi et al., 2004). Additional data are being obtained to validate the current calibrated model.
Numerical results show good agreement with values from the literature for similar lakes. Confidence intervals have also been determined for parameters.
References
Antenucci J. (2000): “The CWR Dynamic Reservoir Simulation Model DYRESM”, User Manual, Centre for Water Research, The University of Western Australia.
Parodi, E. R., Estrada, V., Trobbiani, N., Argañaraz Bonini, G. (2004). Análisis del estado trófico del Embalse Paso de las Piedras (Buenos Aires, Argentina). Ecología en tiempos de cambio. p. 178.
Parodi, E.R., Trobbiani, N., Estrada, V., Argañaraz Bonini, G. (2005) Proliferaciones masivas de Cianoficeas en el Embalse Paso de las Piedras. CONAGUA 2005. (CD).
Rodriguez, M., M. S. Diaz (2006), Dynamic Modelling and optimisation of cryogenic systems, Applied Thermal Engineering, (available online since May 2006).
Shen, J.(2006) Optimal estimation of parameters for aestuarine eutrophication model, Ecological Modelling 191, 521–537.
Zhang, J.J., Jorgensen, S.E., Mahler, H. (2004) Examination of structurally dynamic eutrophication model. Ecol. Model. 173, 313–333.
Presented Tuesday 18, 15:40 to 16:00, in session Environmental Engineering & Management (T1-3).
