Document Type : Research Article
Author
Ferdowsi University of Mashhad
Abstract
Introduction: Actual crop evapotranspiration (Eta) is important in hydrologic modeling and irrigation water management issues. Actual ET depends on an estimation of a water stress index and average soil water at crop root zone, and so depends on a chosen numerical method and adapted time step. During periods with no rainfall and/or irrigation, actual ET can be computed analytically or by using different numerical methods. Overal, there are many factors that influence actual evapotranspiration. These factors are crop potential evapotranspiration, available root zone water content, time step, crop sensitivity, and soil. In this paper different numerical methods are compared for different soil textures and different crops sensitivities.
Materials and Methods: During a specific time step with no rainfall or irrigation, change in soil water content would be equal to evapotranspiration, ET. In this approach, however, deep percolation is generally ignored due to deep water table and negligible unsaturated hydraulic conductivity below rooting depth. This differential equation may be solved analytically or numerically considering different algorithms. We adapted four different numerical methods, as explicit, implicit, and modified Euler, midpoint method, and 3-rd order Heun method to approximate the differential equation. Three general soil types of sand, silt, and clay, and three different crop types of sensitive, moderate, and resistant under Nishaboor plain were used. Standard soil fraction depletion (corresponding to ETc=5 mm.d-1), pstd, below which crop faces water stress is adopted for crop sensitivity. Three values for pstd were considered in this study to cover the common crops in the area, including winter wheat and barley, cotton, alfalfa, sugar beet, saffron, among the others. Based on this parameter, three classes for crop sensitivity was considered, sensitive crops with pstd=0.2, moderate crops with pstd=0.5, and resistive crops with pstd=0.7. Therefore, nine different classes were formed by combination of three crop types and three soil class types. Then, the results of numerical methods were compared to the analytical solution of the soil moisture differential equation as a datum. Three factors (time step, initial soil water content, and maximum evaporation, ETc) were considered as influencing variables.
Results and Discussion: It was clearly shown that as the crops becomes more sensitive, the dependency of Eta to ETc increases. The same is true as the soil becomes fine textured. The results showed that as water stress progress during the time step, relative errors of computed ET by different numerical methods did not depend on initial soil moisture. On overall and irrespective to soil tpe, crop type, and numerical method, relative error increased by increasing time step and/or increasing ETc. On overall, the absolute errors were negative for implicit Euler and third order Heun, while for other methods were positive. There was a systematic trend for relative error, as it increased by sandier soil and/or crop sensitivity. Absolute errors of ET computations decreased with consecutive time steps, which ensures the stability of water balance predictions. It was not possible to prescribe a unique numerical method for considering all variables. For comparing the numerical methods, however, we took the largest relative error corresponding to 10-day time step and ETc equal to 12 mm.d-1, while considered soil and crop types as variable. Explicit Euler was unstable and varied between 40% and 150%. Implicit Euler was robust and its relative error was around 20% for all combinations of soil and crop types. Unstable pattern was governed for modified Euler. The relative error was as low as 10% only for two cases while on overall it ranged between 20% and 100%. Although the relative errors of third order Heun were the smallest among the all methods, its robustness was not as good as implicit Euler method. Excluding one large error of 50%, the average relative errors in this method was less than 10%. However, the ETc is time-dependent and varies from one day to another. So, averaging ETc over a larger time step brings about more error in computations. Accumulated relative error in Eta (ETp=5 mm.d-1, W0=Wj, t=1 d) under medium soil and crop type was decreased as the number of time steps increased, irrespective of the numerical method.
Conclusions: Based on practical considerations, we propose implicit Euler for its robustness, and 3-rd order Heun for its low maximum relative error for all combinations of soil and crop types.
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