Fluid-dynamic and chemical model to evaluate environmental risks in river networks

Pollution of natural river networks is a growing environmental problem which produces more and more alterations to complex and fragile aquatic ecosystems. The main causes of this problem are anthropic emissions of pollutants both as uncontrolled discharge stream and due to accidental sources (i.e. treatment plant failure or chemical disaster in proximity). Integrated numerical models are necessary to evaluate the environmental risk magnitude. These models have to take into account both the natural characteristics of river basin and self purification kinetic. In these present work a numerical model to evaluate the diffusive-convective transport is presented; it analyzes the processes, that occur in river networks, due to an emission of known concentration produced by the river current. The proposed resolutive approach is based on a Lagrangian description of the transport process. As well known (Csanady 1973) this model analyzes the path of a generic solute particle that is characterized, in the case of turbulent flow field, by a random walk. In particular, referring to a generic particle, one can say that the displacement S i, after a time n∆t, is a casual variable and its probability density G ( S i ) is yielded by stochastic equation of Fokker-Planck (Risken 1996). The determination of function G ( x,y,t ) = G ( S i ,t ) is however complex and it is still a subject of specialized mathematical studies (Risken 1996). A remarkable simplification of the problem can be made if one postulates that the flow field is uniform ( U= const., V= const.), the dispersion coefficients constant ( D x = D y = D ) and the solute emission punctual. In that case one can prove (Uffink 1988) that G ( S i ) is the well known expression of Gaussian probability density; the elementary displacement components ∆s x and ∆ s y , during the generic time interval ∆t, are: 1 https://www.w3.org/1998/Math/MathML" display="block"> Δ s x = U Δ t + R 1 6 D Δ t Δ s y = V Δ t + R 2 6 D Δ t in which R is a random number characterized by an uniform distribution into the interval [−1, 1]. In addition, the model is improved to consider the reactivity of the pollutant starting from the simple Steeter’s and Pelps’s approach. The first equation considered in the model states the non-reactive mass balance, to which a consumption term P i, concerning with the oxidation reactions for BOD reduction was to be added. This last term, in the aforesaid simplified model, is: 2 https://www.w3.org/1998/Math/MathML" display="block"> P i = − k C B O D in which k is the chemical kinetic constant. The second equation takes into account both the dissolved oxygen consumption rate, by oxidation reaction, and the oxygen flux crossing the free surface, given by an oxygen deficit. The Steeter and Pelps model assumes that the dissolved oxygen consumption rate can be equal to P i, while the flux is expressed by: 3 https://www.w3.org/1998/Math/MathML" display="block"> N O 2 = β( C O 2 * - C O 2 ) being https://www.w3.org/1998/Math/MathML" display="inline"> C o 2 * the dissolved oxygen concentration in saturation conditions (for a fixed value of temperature) and β an exchange coefficient (Vesilind et al. 1982). This model considers the BOD concentration as representative of the pollutants and postulates a linear dependence between the BOD oxidation kinetics and its concentration. This model allows the evaluation of concentration profiles of pollutant and dissolved oxygen in different hydrodynamic conditions. The hydrodynamic model is monodimensional and stationary and describes various fluvial regimes, such as flood, ordinary and low stream. The chemical model assumes that the BOD and dissolved oxygen concentrations are sufficient to define the whole system from a chemical point of view.