Mathematical Modeling of Filtration from Canals and Sprinklers of Irrigation Systems

We consider several schemes of seepage flows from canals and sprinklers of irrigation systems through the soil layer, underlain good underlying permeable confined aquifer or water-resistant base. For their study and formulated using the method of P.Y. Polubarinova-Kochina solved multivariable mixed boundary value problems of the theory of analytic functions. On the basis of these models are developed algorithms for calculating the size of the saturated zone in situations when the filter has to evaluate the combined effect of the painting movement of such important factors as seepage back pressure from the underlying confined aquifer or an impermeable base, cross-sectional shape and channel the power supply level of water in it , capillarity of the soil and evaporation from the free surface of groundwater. The results of calculations for all the schemes are compared with the same filtration filter parameters depending on the shape of the channel as a power source (canal or irrigation), and the type of foundation soil layer (silnopronitsaemy confined aquifer or aquitard).


Introduction
It is known (see, eg, reviews [1][2][3][4]) that the mathematical modeling of filtration flows from canals and sprinklers of irrigation systems used a variety of research methods applied analytic functions introduced in [5][6][7][8][9], the method of conformal mappings of the rectilinear polygons based on the application of the Schwarz-Christoffel formula [10][11][12][13] and functions Riesenkampf and Verigin [14], the method of P.Y. Polubarinova-Cochina, based on the application of the analytic theory of linear differential equations of Fuchs class [15][16][17][18], the theory of inverse boundary value problems [19,20], etc. Despite the importance of these research studies in this direction so far alienated and not fully reflect the specific movements of the canals and sprinklers, which is characterized by the presence of such important factors as seepage back pressure from the underlying permeable aquifer well containing a pressurized underground (or artesian) water or water-resistant base, soil capillarity and evaporation (or infiltration) at the free surface of groundwater. All marked work associated with the presence of one or two of the three factors that significantly affect the simulated processes. Accounting for the simultaneous influence of these three characteristics of the filter has not become a broad domain of exact analytical solutions.
In contrast to these studies in this paper, which is a continuation and development [21][22][23][24][25], both case of motion of the water of the canals and sprinklers are considered together, which apparently has not yet appeared in the literature. In this case it is shown that the solution for sprinklers obtained by solving the filtration problem for channels with a limit associated with the conformal mapping parameters contained in the required dependencies. Provides a uniform method of solving problems, which allows to consider four different schemes of the flow and take into account in the study at once all three filtration characteristics -capillarity, evaporation, and props -that is of considerable practical interest because it allows us to estimate the combined influence of these factors on the phenomenon. For this purpose, the method of P.Y. Polubarinova-Kochina [1][2][3][4], based on the use of conformal mapping of circular polygons. With the help of developed [26][27][28][29] methods of solving the corresponding linear differential equations of the Fuchs class, which are very common problems of underground hydromechanics solve mixed boundary value problem of multi-parameter theory of analytic functions. On the basis of the obtained exact analytical relationships and numerical calculations are carried out by hydrodynamic analysis of the influence of physical parameters of the schemes of flow on filtration characteristics. We give a comparison of the results of calculations for these schemes under the same filter characteristics depending on the shape of a channel-power sources of water (or irrigation canal) and the type of under-lying soil base (well permeable confined aquifer or aquitard).

Filtration of the Channels in the Presence of the Basis of Well-Permeable Horizon Containing Pressurized Groundwater (Scheme 1)
In fig. 1 shows a right-symmetric picture steady plane flow of an incompressible fluid by Darcy's law with known filtration coefficient к=const of channel width l with a shallow depth of water at uniform evaporation from the free surface of CE intensity ε (0 <ε <1), divided by the coefficient of filtration. At some depth from the bottom of the channel T is silnopronitsaemy pressure horizon FE, piezometric water level in which the 0 k H h + ( k h − static height of capillary rise of groundwater [1,2]) above the horizontal interface between it and a layer of soil, which studies the movement. The channel width l, thickness of the soil layer T, the pressure (above the roof) in the underlying permeable horizon is well along 0 H with the intensity of evaporation ε and high vacuum hk, due to capillary forces in the soil are given. If we introduce the complex potential of motion ω = φ + iψ (φ -velocity potential, ψ -stream function) and complex coordinate z = x + iy, classified кT, and T, then the problem lies in determining the position of the depression curve, CE and, hence, the width k l of the capillary spreading of water filtration and BC flow Q with the following boundary conditions: : : We introduce an auxiliary parametric variable t, the function z(t), conformally maps the half-strip Re 0, t > 0 Im 0.5 t π < < in the t plane, the flow region in accordance points 0, fig. 2, a shows a complex velocity field w , corresponding to the boundary conditions (1), which is a circular quadrangle with a cut to the top of D, and the angle arctg πν ε = at the vertex C. Function effecting the conformal mapping of t on the field w has the form Here 1,2 Y -are linearly independent integrals of the corresponding linear differential equation of Fuchs class [26][27][28], C -a suitable real constant conformal mapping.
Applying the method of P.Y. Polubarinova-Cochina and taking into account the relation (2), as well as the fact that w = dω / dz , we find a parametric solution of (1) as )-unknown ordinates of the points A and F in the plane t [21][22][23]. Integrating the representation (3) along the contour of the auxiliary field t, we can obtain expressions for the physical characteristics of the model Ф , are used to determine the unknown parameters of conformal mappings a, f, C and M. After finding these constants are calculated and the required filtration characteristics k l and Q coordinates of points of depression 0 1 , In (4) -(6) the integrands -right-hand sides of expression (3) in the corresponding sections of the contour plane t.
Private and limiting cases of figure 1 are described in detail in [23].

Filtration of the Irrigators in the Presence of the Basis of Well-Permeable Horizon Containing Pressurized Groundwater (Scheme 2)
In fig. 1 shows the pattern of motion of the sprinkler borozdovogo type with a radius l which is filled with water up to ground level, taken as a horizontal plane. During the initial review sprinkler replace a point source located at a point A . Then the solution for the source is obtained by solving the problem of filtration for the channel (3) with the values of conformal mapping 0.5 , a π = 1, A = is at the confluence of the points A and B in the plane of flow.
Further, according to [1,2,6,7], we take one of the lines of equal pressures, for example BG (shown in fig. 1 smooth curve) for the cross-section of the sprinklers with radius l, and let her 0.

Comparison of Numerical Results for the Channels (Scheme 1) and Sprinklers (Scheme 2)
In fig. 1 shows the curves of depression during the filtration of the channels (left curve CE) and sprinklers (right curve CE), calculated with 0.05, Analysis of the data tables and graphs to the following conclusions.
The increase in channel width and radius of the sprinkler, the pressure in the underlying aquifer and the height of the vacuum caused by capillary forces in the soil and reduce evaporation and capacity of the soil layer lead to increase the width of the capillary spreading of the water. Reduce evaporation and increase the backwater from the waters of the underlying horizon is accompanied by increase in the magnitude k l and, conversely, decrease the filtration rate Q . A similar behavior is observed k l and Q the values and decreasing capacity of the soil layer T. Thus, with respect to the filtration of the canals and sprinklers and evaporation capacity of the soil layer plays the same role as props, as the quantity 0 H .   The last block of table 2 makes it possible to analyze the impact of cross-sectional shape channel power sources, comparing the results for the channels (scheme 1) and sprinklers (scheme 2) with the same filter characteristics. A comparison of these results, as well as from the graphs in scheme 2, it follows that if sprinklers are the final specifications k l and Q more than in the case of channels, with the difference for these values of 1.4-2.2 and 1.3-1.5 times, respectively.

Filtration of the Channels in the Presence of Water Pressure at the Base (Scheme 3).
As discussed in section 2 − 4 schemes 1 and 2, underlying soil layer underlying confined aquifer of high permeability is an important factor, by which the drainage of the soil can be carried out not only with free filtering, but also in the presence of backwater. In those situations where the soil layer is underlain by an impermeable base, the only way to tap filtered water channel is the evaporation from the free surface of ground water. In such cases, naturally raises the question of determining the size of the zone of saturation in the filter channel.
In fig. 4 shows the pattern of motion of the channel AB in a layer of soil capacity T, underlain horizontal aquitard FE. As an auxiliary parametric variable, it is convenient to choose the previous half-strip plane t , but with a different line of points 0, In this case, the complex velocity field corresponding to the boundary conditions (1), (8), is a circular pentagon with the same incision with apex at point D, but with the two angles at the vertices of E and C, are equal (1 ) π ν − and 2arctg πν ε = , respectively ( fig. 2, b). Function giving a conformal mapping of the parametric variable t in the region w has the form Applying the method of P.Y. Polubarinova-Kochina and proceeding as above, using the relation (10) we obtain the dependences and filtration rate Q using the formula (9). In the limiting case of motion where there is no soil capillarity, at h k = 0 parameters с = 0, B = 1, C = ∞ and the results obtained [16].
In fig. 4 shows the flow pattern of the channel, calculated with 0.4,  First of all, attention is drawn to the same qualitative dependence on the parameters k l and Q variables, l , k h and ε : increasing the channel width, the static height of capillary rise of water, as well as reducing evaporation and bed thickness lead to higher values k l . An increase in the size of the zone of saturation by increasing the spreading of liquid on the water pressure L is related to figure 3 to increase capacity of the soil layer, which is quite natural from a physical point of view.
The data in table 3

Filtration of Water-Filled Channels in the Presence of Water Pressure at the Base (Scheme 4)
Up to now study the motion of the channels with shallow depth of water. Designed to study the complex analytical scheme 3 (formula (11) (13)) allows us to consider the channels that are filled with water, and thereby estimate the combined effect of the depth of water in canals and evaporation from the free surface of the water table, but in the absence of capillarity, soil, at In fig. 5 shows the flow pattern of the AB channel of rectangular cross section width l of the water depth H .
In this scheme the boundary conditions (1), (8) in areas AF, CE and FE are stored, and the conditions on the segments AB and BC are replaced by these: : where H ( 0 H T ≤ < ) -depth of water in the channel. Taking into account the coincidence of the complex velocity to that of scheme 3 ( fig. 2, b) After finding the unknown constants of conformal mapping of a, b and M determine the width of the zone of saturation L is calculated filtration flow rate Q by the formula (9), taking into account the fact that in this scheme 0. ε l and T is given in [30]. Below focus on comparing the simulation results for circuits 3 and 4. In table. 4 summarizes the results of calculations of the influence of physical parameters of the scheme on the width of the spreading of water in water-resistant base L and consumption Q .  The analysis shows that in both schemes 3 and 4, reducing evaporation and increasing the capacity of the soil layer and the width of the channel leading to an increase of the saturated zone. Evaporation continues to have a very significant impact on the spreading width of the water: according to the table 4 with decreasing width parameter increases by 2.2 times. As previously in figure 3 the dependence L of the channel width l is close to linear, and depending on the quantities L and Q the water level H are qualitatively similar if we bear in mind that (9).
With regard to flow , Q in both schemes, with an increase in performance , ε l and T consumption increases, with the largest influence on the reservoir Q has a capacity T : from table 4 shows that the variation of the parameter T is accompanied by an increase of 313%, ie practically the same as in figure 3. However, in comparison with the scheme 3 biggest changes by varying the layer thickness is undergoing now spreading width of the water: with increasing T width L parameter increased by almost 200%.
The data in table 4 that the parameter H leads to a very small (in the range of 20-30%) deviations of the width of the spreading of water L and the seepage flow, so that the influence of water level in the canals of little practical effect on the flow pattern.

Conclusions
Developed in the computational scheme of filtering channels and sprinklers of irrigation systems in soils underlain silnopronitsaemym confined aquifer or water-resistant base, allow us to make a comparative assessment of the role of cross-sectional shape channel power supply and the water level in it, backwater from the waters of the underlying aquifer or confining layer, soil capillarity and evaporation or seepage at the free surface of the ground water flow pattern.
In particular, we establish the following: − compared with the filtration of the sprinklers for irrigators width of the capillary spreading of the liquid and the magnitude of the seepage flow can exceed the corresponding characteristics of more than 50%; − the greatest influence on the width of the capillary spreading of water and filtration rate has capillarity of soil, which improves the efficiency of canals and irrigation canals; − influence the level of water in the channels has practically no effect on the size of the zone of saturation and value of the seepage flow.