Abstract
This work arises from the need of exploring new features for modeling and optimizing water consumption in irrigation processes. In particular, we focus on water flow model in unsaturated soils, accounting also for a root water uptake term, which is assumed to be discontinuos in the state variable. We investigate the possibility of accomplishing such optimization by computing the steady solutions of a theta-based Richards equation revised as equilibrium points of the ODEs system resulting from a numerical semi-dicretization in the space; after such semi-discretization, these equilibrium points are computed exactly as the solutions of a linear system of algebraic equations: the case in which the equilibrium lies on the threshold for the uptake term is of particular interest, since the system considerably simplifies. In this framework, the problem of minimizing the water waste below the root level is investigated. Numerical simulations are provided for representing the obtained results.Article HighlightsRoot water uptake is modelled in a Richards’ equation framework with a discontinuoussink term.After a proper semidiscretization in space, equilibrium points of the resultingnonlinear ODE system are computed exactly.The proposed approach simplifies a control problem for optimizing water consumption.
Highlights
Introduction to the Physical ProblemSoil water content changes across the unsaturated zone because of inflows of water via infiltration from the surface, due, e.g., to rainfall and irrigation, and outflows caused by evaporation and root water uptake as well as percolation beneath the root zone
2 The Proposed Model In Jarvis (1989), an empirical model is used, which assumes that the soil depth is split into different layers, and water uptake is given as a function both of the potential transpiration rate and of a weighted stress index which accounts for the effects of the vertical distributions of roots and soil water content, for each layer
Once we have evaluated the equilibrium points i for our system (11), it is necessary to discuss for which choice of boundary conditions ( 0, N) and thresholds θi, such solution can really describe a realistic equilibrium in the chosen configuration
Summary
Soil water content changes across the unsaturated zone because of inflows of water via infiltration from the surface, due, e.g., to rainfall and irrigation, and outflows caused by evaporation and root water uptake as well as percolation beneath the root zone. This study has been introduced in Warrick (1974), where the steady-state assumptions are justified as an approximation of high-frequency irrigation: a semi-infinite column or a saturated bottom boundary conditions are therein considered, together with different extraction functions Such interest is confirmed in the milestone paper Philip (1989), motivated since, even for multidimensional unsaturated flows, the steady solutions are approached rapidly if close to the water source term, concluding that the application interest for such investigations is more practical than it might seem at first glance; on the other hand, the advantage of handling just linear systems is therein highlighted.
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