Simulation of the movement of shallow water along furrows during surface irrigation

. The article presents a mathematical model of the movement of irrigation water along furrows. This work is aimed at determining the movement of water and the possibility of correctly adjusting its movement. Based on the above conditions, this article provides a more correct description of the dynamics of surface waters, necessary to take into account dispersion effects.


Introduction
Non-pressure filtration flows with a free surface on which the liquid pressure is constant and equal to external atmospheric pressure are characteristic of the filtration of irrigation water through soil, caused by furrow irrigation of cotton.
The problems of pressure-free filtration of unsteady one-dimensional flows of incompressible fluid, including unsteady flow regimes under linear and nonlinear filtration laws, have not been sufficiently studied.This is due to the complexity of their mathematical description and obtaining the shape of the depression surface.

Materials and methods
For a mathematical description of the movement of irrigation water along furrows, there are various models.The choice of model depends on the class of problems being solved.In practice, the shallow water model (Saint-Venant equations) is widely used.
The derivation of shallow water equations is based on the hydrostatic approximation [1][2].Let us consider a layer of irrigation water in a furrow, the depth of which is H, the bottom topography is described by the function the free surface level is equal to , while the specific gravity vector is directed opposite to the z axis, where -is the acceleration of free fall.The fluid flow velocity v has three components u and , along the axes, respectively.Then, to describe such a layer of irrigation water along the furrows within the framework of the theory of shallow water, it is necessary to fulfill the following conditions where L -is the characteristic spatial scale of heterogeneity in the horizontal direction,is the characteristic time of the processes under study, is the correction factor.
For a more correct description of the dynamics of surface waters, it is necessary to take into account dispersion effects, going beyond the hydrostatic approximation [14,16].Various approaches are being considered that take into account the effects of vertical motion in different approximations [7][8][9][10][11][12][13][14][15].
Based on the laws of conservation of mass and momentum of a homogeneous incompressible fluid with density ρ=const, we write the equations of hydrodynamics in integral form for arbitrary volumes of fluid ("liquid particles"). ( ( Where -is isotropic pressure, -is the fluid velocity in the plane, -is the density of external volume forces in the horizontal direction, -is the surface density of sources and sinks liquid [m/s], -is the cross-sectional area of the "liquid particle" in the plane In shallow water theory, the vertical component of velocity w is neglected, since it is assumed that | .Then, taking into account this assumption, equation (3) can be represented as a hydrostatic balance equation.(4) Where -is atmospheric pressure.Thus, equations ( 1), ( 2) can be written as

Where
are the average values over the z coordinate of the velocity and density of body forces in the plane (x,y), respectively.Let us apply the rule for differentiating integrals with respect to time to integral equations:

Where
-is an arbitrary tensor function.As a result, we obtain the shallow water equations in differential form (Saint-Venant equations):

Where
-are the components of the water velocity vector in the source or sink V.In the general case, the value of σ(x,y,t) is determined by the work of various types of hydraulic structures (σ>0 -source, σ<0 -drain), precipitation, evaporation, infiltration.In our models, the most important characteristic is the hydrograph with the rate of water inflow Q (m 3 /s); as a rule, we specify a spatially distributed source in a certain area (for example, behind a hydroelectric dam) and the hydrograph is determined by the integral over the area The model takes into account the action of the following specific forces (11) Where -is the quadratic model of bottom friction, λ -is the coefficient of hydraulic resistance, -is the Coriolis force, is the angular the speed of rotation of the Earth, is the force due to the presence of wind, -is the force of viscous friction.The coefficient λ is a phenomenological characteristic, which in turn depends on other parameters that are poorly determined from theory, and various options are considered in the literature, which significantly depend on the formulation of the problem [13][14].Various models are known in the literature for specifying hydrological resistance to flow [14].All of these models are phenomenological, since they attempt to reduce a large number of complex, often poorly formalized within the framework of a two-dimensional approximation, physical factors to simple algebraic relationships that connect resistance with the geometry of the channel, the average velocity of the fluid, the integral properties of the underlying surface (bottom inhomogeneities at different scales), laminar/turbulent nature of the flow (turbulent viscosity), hydraulic slope and possibly other characteristics.This approach appears quite often in research in various fields.In the case of a shallow water model, the role of such a parameter is played by the Manning coefficient, the Chezy coefficient, the relative roughness parameter, the hydraulic roughness coefficient, or some other similar parameter, which are often related.Their quantitative assessment is completely determined by observational or experimental data.Recently, in connection with the achievements of numerical experiments, it has become possible to make such estimates based on direct hydrodynamic modeling.In [13][14], an attempt was made to obtain such estimates by determining the relationship between hydrological resistance and the characteristics of the underlying bottom surface.
We will limit ourselves to considering the formula traditional for numerical models of shallow water dynamics [1,11,18] (12) , 04047 (2024) E3S Web of Conferences https://doi.org/10.1051/e3sconf/202449404047494 AEES2023 Where -is the bottom roughness coefficient according to Manning (we will call it the roughness coefficient or Manning coefficient).It should be noted that the value n_M has a complex dimension , but according to tradition, it is not indicated explicitly, meaning, for example, that .The specific friction force between the air flow in the surface layer of the atmosphere (wind) and the liquid flow is determined by the nonlinear relationship [2,16] (13 Where -is a parameter depending on the state of the water surface, and -are the density of air and water, respectively, is the wind speed vector in the horizontal direction.In (13) we take into account the movement of the fluid with speed .As a simple model for the parameter , we can use a linear dependence on the relative wind speed Where the speed is specified in [m/s].
The results of studying the interaction of wind fields in the atmosphere and surface movements in the world ocean are considered in [9,12].
To calculate the viscous force in the case of a non-uniform viscosity coefficient we can write (15) Where . .

Results and Discussion
The issue of choosing the parameter ν is actively discussed in the scientific literature [3,14].It should be noted that the value ν is turbulent viscosity, phenomologically describing developed turbulence, for which, following the Landau model are the characteristic speed and size of turbulent pulsations, respectively).The quantity , and the speed is determined by the speed of propagation of short surface gravitational waves .To describe processes on scales <Δx, subgrid physics methods are used.It is important to note that the practice of using this kind of models indicates the need to take into account spatial heterogeneity in addition to the bottom function and also the quantities .These functions are specified as matrices on a numerical grid .Of particular importance is the heterogeneity of the Manning coefficient, since its value depends quite strongly on the properties of the underlying surface [5,16].Moreover, there are indications that the value of may depend for a significantly unsteady flow on time, parametric through the thickness of the liquid layer H(t).
The factor of water loss from the surface layer due to evaporation and infiltration can be significant in some problems.In the simplest case, we can limit ourselves to specifying constant , or spatially inhomogeneous distributions for the study area depending on the properties of the underlying surface or wind regime, air and water temperatures, and the surrounding landscape (irrigated field).
, 04047 (2024) E3S Web of Conferences https://doi.org/10.1051/e3sconf/202449404047494 AEES2023 However, although this approach seems more accurate than specifying a constant, it is uncontrollable because it does not take into account the time dependence.
Modeling the interaction with groundwater is more correct [1,16,18].Let's write a simple model, the so-called nonlinear saturation model, which better reproduces experimental data than traditional linear or nonlinear filtering models.
Assuming that the value is proportional to the thickness of the liquid layer, we write (16) Where determines the rate of absorption of a layer of water of thickness H into dry soil, the value takes into account the nonlinearity of this process, providing a saturation effect.In the simplest case we can assume that (17) Where is the initial thickness, and -is the soil moisture saturation coefficient, determined through the ratio of the volume of water , contained in soil with a volume of -is the soil moisture saturation coefficient.As filtration proceeds, the value of α increases, but the rate of this growth should decrease so that at long times → 0.
When there is no water in the soil, the filtration rate is determined by the source , the characteristic thickness of the soil , inside which water is distributed, and soil porosity .As a result, we write the equation for the change in coefficient α: (18) Where the characteristic time determines the time of soil drying due to either evaporation, if the surface water has left, or due to infiltration into deeper layers, which are characterized by less porosity.

Conclusions
In this work, based on a unified approach, the defining equations of the unsteady low water equation along a furrow with an unsteady bottom are obtained for the case of a deforming or moving bottom.To derive the equations, it was assumed that the vertical component of the current velocity is a linear function of the z coordinate, and the velocity components in the horizontal plane do not depend on z at all.And also, the velocity components in the horizontal plane are considered quadratic functions of z, and the potentiality of the flow and the different scale of processes in the vertical and horizontal plane are assumed.