Border Irrigation Modeling with the Barré de Saint-Venant and Green and Ampt Equations

Abstract

In gravity irrigation, how water is distributed in the soil profile makes it necessary to study and develop methodologies to model the process of water infiltration and redistribution. In this work, a model is shown to simulate the advancing front in border irrigation based on the one dimensional equations of Barré de Saint-Venant for the surface flow and the equation of Green and Ampt for the flow in a porous medium. The solutions were obtained numerically using a finite difference Lagrangian scheme for the surface flow and the Raphson method for the subsurface flow. The model was validated with data obtained from the literature from an irrigation test and its predictive capacity was compared with another model and showed excellent results. The hydrodynamic parameters of the soil, necessary to obtain the optimal irrigation discharge, were obtained through the solution of the inverse problem using the Levenberg–Marquardt optimization algorithm. Finally, the results found here allow us to recommend that this model be used to design and model border irrigation, since the infiltration equation uses characteristic parameters of the physical soil.

How Surface Irrigation Contributes to Climate Change Resilience—A Case Study of Practices in Mexico

“…With the soil characterization data, initial water content, inflow discharge, and the results from the irrigation test (advance and recession time), we proceeded to obtain the characteristic parameters of the infiltration equation that represent the irrigation event. The water movement simulation over the soil surface is described by the Barré de Saint-Venant equations [39]:…”

Hydrodynamic Border Irrigation Model: Comparison of Infiltration Equations

“…where h is the water depth, q(x,t) = U(x,t)h(x,t) is the discharge per unit width of the border or the unitary discharge, x is the spatial coordinate in the main direction of the water movement in the border, t is the time, U is the mean velocity, β = U IX /U is a dimensionless parameter where U IX is the projection in the direction of the output velocity of the water mass due to the infiltration, V I = ∂I(x,t)/∂t is the infiltration flow, that is, the water volume infiltrated per unit width per unit length of the border, I is the infiltrated depth, g is gravitational acceleration, J o is the topographic slope, and J is the friction slope that can be determined by the fractal law of hydraulic resistance [3]:…”

“…For the discretization of Equations ( 1) and ( 2), a finite-difference Lagrangian scheme was used for its solution [3,17]. Figure 1 shows the flows and depths of water on the surface as infiltrates at times t i and t i+1 with the subscripts J, M, L, and R to identify the time step and the boundary conditions in a specific cell.…”

“…In surface irrigation, the water is distributed throughout the soil profile, making it necessary to study and develop methodologies to model the water infiltration and redistribution processes [3]. The infiltration process varies over time, and with the characteristics of the plot, it affects the surface's flow by making it unstable and spatially variable [4].…”

“…Software such as SIR-MOD [13] and SISCO [14] has been developed through a finite difference scheme of rectangular cells and, for the subsurface flow of water, they use the Kostiakov-Lewis equation, while other models use the Kostiakov equation and a solution method based on an explicit algorithm with a MacCormack scheme [15]. Other models use infiltration equations that are based on physically based parameters, such as that of Green and Ampt [3,16], where the moisture profile is assumed to be constant, and the models that incorporate an analytical solution for the infiltration in a soil with a shallow water table for the advance phase of border irrigation [17].…”