Analysis of Concentration as Sampled in

The aim of this paper is to analyze the statistical properties of solute concentration in natural aquifers as sampled in observation wells, having a small diameter in comparison with the characteristic size of the heterogeneity in hydraulic properties. The analysis, in Langragian framework, takes advantage of the reverse formulation, where, instead of considering the destination of the injected particles, the origin of the particle being sampled is sought. In the case of small values of the log-conductivity variance _Y², it allows the derivation of an analytical expression for concentration mean, variance and pdf, while for aquifer characterized by high value in _Y², a numerical analysis based on a Monte Carlo approach using a reverse scheme is developed and applied for values of _Y² up to 2. In this case, the use of a Beta function to fit the concentration pdf proves valid for practical applications. The comparison between the numerical and the analytical results defines the range of validity of the analytical ones. The relative role of large-scale dispersion processes and pore-scale effects is analyzed in terms of global variance in order to point out limits and accuracy of the Eulerian scheme in comparison with the Lagrangian one.     

Reformulation and a Lagrangian heuristic for lot sizing problem on parallel machines

We consider the capacitated lot sizing problem with multiple items, setup time and unrelated parallel machines. The aim of the article is to develop a Lagrangian heuristic to obtain good solutions to this problem and good lower bounds to certify the quality of solutions. Based on a strong reformulation of the problem as a shortest path problem, the Lagrangian relaxation is applied to the demand constraints (flow constraint) and the relaxed problem is decomposed per period and per machine. The subgradient optimization method is used to update the Lagrangian multipliers. A primal heuristic, based on transfers of production, is designed to generate feasible solutions (upper bounds). Computational results using data from the literature are presented and show that our method is efficient, produces lower bounds of good quality and competitive upper bounds, when compared with the bounds produced by another method from the literature and by high-performance MIP software.     

Solute Concentration Statistics in Heterogeneous Aquifers for Finite Péclet Values

A Lagrangian framework is used for analysing the concentration fields associated with transport of nonreactive solutes in heterogeneous aquifers. This is related to two components: advection by the random velocity field v(x) and pore-scale dispersion, characterized by the dispersion tensor D _d; the relative effect of the two components is quantified by the Péclet number. The principal aim of this paper is to define the probability density function (pdf) of a nonreactive solute concentration and its relevant moments >C< and ² _c as sampled on finite detection volumes. This problem could be relevant in technical applications such as risk analysis, field monitoring and pollution control. A method to compute the concentration statistical moments and pdf is developed in the paper on the basis of the reverse formulation widely adopted to study solute dispersion in turbulent flows. The main advantages of this approach are: (i) a closed form solution for concentration mean and variance is attained, in case of small size of the sampling volume; (ii) a numerically efficient estimate of the concentration pdf can be derived. The relative effects of injection and sampling volume size and Péclet number on concentration statistics are assessed. The analysis points out that the concentration pdf can be reasonably fitted by the beta function. These results are suitable to be employed in practical applications, when the estimate of probability related to concentration thresholds is required.