Keywords: mathematical modelling, convection-diffusion-reaction, stabilized finite element, residual-free bubbles, pollutants leaching.

Pesticides used in crop production and herbicides used for weed control are the major source of nonpoint-source pollutants to groundwater. Their discharge to the surface water may be a contributing factor toward the decline of the living resources and the deterioration of ecosystems. In fact, the raising presence of this kind of pollutants in the aquifers with high concentrations has become a serious environmental problem.

Physically-based environmental simulation models can be cost-effective tools as an alternative to costly and prolonged field monitoring strategies. Environmental fate and transport simulation models vary in their complexities and purposes. The more complex the model is, the more comprehensive in the level of detail, but their use, in practice, may be hampered by lack of sufficient data that justify their complexity.

Leaching of pollutants (pesticides, herbicides, etc.) in soils and groundwater is a complex process. Agricultural practices, soils properties, climatic conditions are some of the most important aspects to be considered. Several simulation models, more or less complex, have been developed and used by several authors in order to simulate the behaviour of soil and aquifers for the chosen pollutants (see [1] and [2] as examples). A summary of several models and its theoretical basis can be found in [3] and [4].

The two models studied in this paper are a linear equilibrium and a linear non equilibrium adsorption models based on the one-dimensional convection diffusion equation. The models account for processes such as diffusive mass transfer, biochemical degradation, crop uptake, volatilization, linear equilibrium adsorption and gravity drainage.

The mathematical problem to be solved is an unsteady linear convection-diffusion-reaction one-dimensional problem. The use of a numerical method to approximate the solution this problem allows to solve different problems modifying the initial, boundary and physical conditions, and it is a first step for future works dealing with more realistic multidimensional problems.

Most of the previous works on leaching of pollutants simulations use analytical solutions of much simpler models. As an example, in [5] integrated mass-balance equations are used to approximate the average solute concentration on each zone (root-zone, vadose-zone and groundwater) and then, the corresponding time-dependent first order differential equation is analytically solved. The advantages of using a numerical approximation is that it may be generalized in a straightforward way to the case of different zones without simplifying assumptions.

The main difficulty in the numerical approximation of this kind of partial differential equations, is the accurate modelling of the interaction between convective, diffusion and reaction processes. In this work we consider several stabilization techniques, including also the Link-Cutting Bubble strategy recently proposed in [6]. As it will be shown by means of numerical experiments, this stabilization technique is the one that produces better results when approximating the problems derived from the models developed in this work.

1

J. González and L. Ukrainczyk, "Transport of Nicosulfuron in Soil Columns", J. Environ. Qual., 28, 101-107, 1999.

2

L. Candela and M.A. Mariño, "Simulation of 2,4-D herbicide transport through the unsaturated zone using an analytical model", Internationa J. Environ. Analytical Chemistry, 84(1-3), 123-131, 2004. doi:10.1080/03067310310001593756

3

M.M. Hantush, M.A. Mariño, M.R. Islam, "Models for leaching of pesticides in soils and groundwater", J. Hydrology, 227, 66-83, 2000. doi:10.1016/S0022-1694(99)00166-3

4

A. Vincent, Impact de l'implantation de bandes enherbées sur le transport d'herbicides: étude sur colonnes de sol non perturbé, DEA Hydrologie, Géostatistique et Géochimie, Université Pierre et Marie Curie, Paris, 2003.

5

M.M. Hantush, M.A. Mariño, M.R. Islam, "An analytical model for the assessment of pesticide exposure levels in soils and groundwater", Envir. Mod. & Assess., 1, 263-276, 1996. doi:10.1007/BF01872154

6

F. Brezzi, G. Hauke, L. D. Marini, and G. Sangalli, Link-cutting bubbles for the stabilization of convection-diffusion-reaction problems, Math. Models Methods Appl. Sci., 13 (2003), pp. 445-461.
Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday. doi:10.1142/S0218202503002581

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