Princípio da máxima entropia aplicado à modelagem hidrodinâmica e à eficiência de retenção de sedimentos em pequenos reservatórios

Abstract

Given the need to develop a model that takes into account the uncertainties inherent to the physical processes that involve the flow of water and sediment and that improves the results obtained by traditional deterministic models, this work used the principle of maximum entropy according to Shannon. The entropy concept, derived from the Information Theory, supported our approach in the evaluation of hydrodynamic and sedimentological models. In the first stage of the research, a relevant scientific question was assessed, namely, what gain of information there is with the increase of constraints and the robustness of primary formulation, which are part of the principle of maximum entropy. For that, the selected problem to be tackled was the velocity field in open channels, fundamental in the study of the water flow. We examined the trade-off between the number of constraints (one, two or three) and the premise (or primary formulation: weak, using Cartesian coordinates or strong, using curvilinear coordinates). We evaluated six hydrodynamic entropy models that generate velocity fields in open channels. Five of these models were obtained from the literature and one was developed within the scope of the research. The performance of the models was assessed using the Nash-Sutcliffe coefficient and the root mean square error. For this purpose, we used 1730 accurate velocity measurements made in laboratory by Laser-Doppler velocimetry. We concluded that the models with two restrictions performed better than those with one restriction, due to the gain of information inserted into the model by the second restriction. We also observed that models with three restrictions performed worse than those with two restrictions. The gain of information by adding the third constraint was overcompensated by the degradation of information caused by the numerical solution, since there is no algebraic solution for more than two restrictions. The use of curvilinear coordinates represents the velocity field more accurately. We observed that the models that used the strong premise (curvilinear coordinates) presented better results than the models with weak premise (Cartesian coordinates). The hydrodynamic model with the best performance (denominated U2ξ) has two restrictions and the strong premise, that is, it uses curvilinear coordinates. The results show, therefore, that there was more gain of information with the application of a strong premise than with the insertion of a third constraint. The two main results of the first stage were useful in the elaboration of the second stage, whose objective was to improve a model designed to assess the trap efficiency of sediment in small dams: (1) for the application of the principle of maximum entropy, we used two restrictions coupled with a strong premise; and (2) the hydrodynamic routine uses the Prandtl-von Kàrman logarithmic model, which corresponds to the one-dimensional U2ξ model. Considering that the Brazilian north-eastern semiarid region has tens of thousands of ungauged small dams, we chose the Camp model as a reference to calculate the sediment trap efficiency. Despite its merit, the Camp model is deterministic and does not admit the sediment resuspension. Thus, this research introduced the probability of sediment resuspension in the model, calculated by the principle of maximum entropy. The probability of sediment resuspension was modelled using two constraints and taking into account the balance of forces (weight, buoyancy, hydrodynamic and cohesive forces) that act on a sediment particle deposited in the reservoir bed. The resuspension probability was then inserted in the Camp equation, generating the Retsed model, which estimates the efficiency of sediment retention in small reservoirs. The simulations by both models were compared to the measured value in the field, using statistical tests, scatter plots and boxplots. We concluded that the Retsed model has performed satisfactorily (error smaller than 6%) and that it did not mimic the systematic error of the Camp model, that is, it did not overestimate the sediment retention because it assumes the resuspension of sediment.