In Figure 1 the rectangular crossing point with slopes of AA₁ and DB on the impenetrable horizontal basis of length of L is presented. Water height in the top tail ofН, lower tail with water level of Н₂, having partially impenetrable vertical wall CD(screen), adjoins a layer sole. If the working part of the crossing point CB (filter) of width ofH₁ is flooded, H₂>H₁, an interval of seepage, usual for dams, is absent 1. The upper bound of area of the movement is the free surface of AD, coming to the disproportionate CD, screen to which there is a uniform evaporation of intensity ε (0 < ε < 1). Soil is considered uniform and isotropic, the current of liquid submits to Darci law with known coefficient of a filtration κ = const

We will enter the complex potential of the movement ω= φ+iψ (φ – speed potential, , ψ – function of current) and complex coordinates  z=x+iy, carried respectively κH and H, where H – a pressure in A point.

At choice of system of coordinates specified Figure 1 and at combination of the plane of comparison of pressures with the  y=0 plane on border of area of a filtration the following regional conditions are satisfied:

(1)

The task consists in definition of provision of a free surface of AD and finding of ordinate of H₀ – points of an exit of a curve depression to the impenetrable screen, and also a filtrational expense of Q.

For the solution of a task we use P. Y. Polubarinova-Kochina's method which is based on application of the analytical theory of the linear differential equations of a class of Fuchs 12345620. We will enter: auxiliary area t – semi-strip Ret > 0, 0 < Imt < 0.5π a parametrical variable t at compliance of points t_A =∞, t _A1 = arcth+0.5π, t_B =arcth+0.5πi (1 < a₁ < b < ∞), a₁, b – unknown affixes of points A₁ and B  in the plane t, t_C =0.5πi and t_D =0; function z (t), conformally displaying a plane t semi-strip on area z, and also derivative dω / dt и dz / dt.

We will address to area of complex speed of  w, corresponding to boundary conditions (1) which is represented a circular quadrangle of ACDE with a section with top in E point (the corresponding inflection point of a

curve depression) and a corner at A, top belongs to a class of polygons in polar grids and was investigated 1213141516171819 earlier. It is important to emphasize that similar areas, despite the private look, however are very typical and characteristic for many problems of an underground hydromechanics: at a filtration from channels, sprinklers and reservoirs, at currents of fresh waters over based salty, in problems of a flow of the tongue of Zhukovsky in the presence of salty retaining waters (see, for example, 921).

The function making conformal display of a semi-strip to area of complex speed of w, has a former appearance 9

(2)

where С (C1) – some suitable material constant.

Defining characteristic indicators of the  dω / dt and dz / dt functions about regular special points 12345620, considering that  w =dω / dz and in view of a ratio (2), we will come to dependences

(3)

where М>0 – a large-scale constant of modeling.

It is possible to check that functions (3) meet the boundary conditions (1) reformulated in terms of the dω / dt и dz / dt, functions     and, thus, are the parametrical solution of an initial regional task. Record of representations (3) for different sites of border of a semi-strip with the subsequent integration on all contour of auxiliary area of the parametrical t leads to short circuit of area of a current and, thereby, serves as control of calculations.

As a result we receive expressions for the set sizes: width of the L crossing point, water level in the top H and the lower H₂the tail`s and lengths of H₁ of the filter

(4)

and also required coordinates of points of a free surface AD

(5)

and expressions for a filtrational expense of Q and ordinate of a point of an exit of a free surface to the screen

(6)

Control of the account are other expressions for sizes  Q,H₀ and L

,(7)

(8)

(9)

and also expression

(10)

directly following from boundary conditions  (1).

In formulas (4)  – (10) subintegral functions – expressions of the right parts of equalities (3) on the corresponding sites of a contour of auxiliary area t.

Limit case. At merge of points of A and A₁, in the plane t, at a₁→1 (arcth a₁ = ∞) the crossing point degenerates in free-flow layer semi-infinite at the left and the task about a current of ground waters to imperfect gallery investigated earlier 9 turns out.

Representations (3) – (10)contain four unknown constants of M, C, a₁ and b. The parameters a₁, b (1< a₁ < b < ∞), C (C ≠  1) are defined from the equations (4) for the set sizes H₁, H₂ (H₁ ≤ H₂ < H) and L, constant modeling of M thus is from the second equation  (4), fixing water level H in the top tail of a crossing point. After definition of unknown constants consistently there is a filtrational expense ofQ ordinate ofH₀ of a point of an exit of a curve depression to an impenetrable site  DC on formulas (6) and coordinates of points of a free surface of DA on formulas  (5).

In Figure 1 the current picture calculated at  ε = 0.5 , H = 3, L =2, H₁ = 1.0, H₂ = 1.4 (basic option 9)  is represented. Results of calculations of influence of the defining physical parameters ε, H, H₁, H₂ and L at sizes Q and H₀ are given  in Table 1, Table 2. In figure 2 dependences of an expense ofQ (curves 1) and ordinates H₀ of an exit of a curve depression to the screen (curves 2) from parameters ε, H, H₁, H₂ and L.

The analysis of these tables and schedules allows to draw the following conclusions.

First of all opposite qualitative nature of change of the sizes Q and H₀ at a variation of parameters attracts attention ε, H and L (Table 1):  also, as well as earlier 9 reduction ε and increase H is led to increase of an expense and ordinates of an exit of a curve depression to the screen. Thus, in relation to a filtration in a crossing point reduction of intensity and evaporation plays the same role, as well as increase in a pressure. Thus the greatest influence on the sizes Q and H₀ renders a pressure: at increase of parameter H by  only 1.2 times the expense and ordinate increase more, than 52 and 24% respectively.

Essential interest is represented by dependences of an expense of a crossing point and ordinate of a point of an exit of a free surface to the screen from water level of H₂ in the lower tail, and also from extent of deepening of the screen, i.e. from the size H₁ at fixed ε, H and L (Table 2). Here as well as concerning parameters ε and H observed opposite qualitative nature of change of the sizes Q and H₀ at a variation of H₁ and H₂. It is visible that increase in water level of H₂ in the lower tail and reduction of deepening of the H₁ screen are followed by reduction of an expense and raising of a free surface that, in turn, it is expressed in increase in H₀; both of these factors characterize strengthening a subtime.

Follows from Table 1 and Figure 2 that reduction of the  H₁ и H₂ parameters respectively at 1.45 and 1.29 times attracts change of size Q for 16.8 % (at fixation of H₁) and 12 % (at fixation of H₂). Noted regularities lead to the conclusion that the expense of a crossing point depends on the size of lowering of the level in a little bigger degree, than on filter length (or from imperfection of a well or a well).

From Figure 2 it is visible that for basic option almost all dependences of the sizes Q and H₀ on parameters ε, H, H₁, H₂ and L are close to the linear.

Comparison of the results received for basic option Q =1.155 and H₀=1.776 with results Q =1.141  and H₀=1.768 for basic option 9 where the current area was limited equipotential at the left shows that the relative error is very small and makes only 0.5 and 1.3% respectively.

Comparison of value of the expense Q =1.16, received for basic option to Q =1.26, value which follows at application of the generalized I.A. Charny's formula 1 ^{with. 267} for a usual rectangular crossing point (without screen) in the presence of evaporation

leads 8.3% to an error.

For comparison with results 7 we will consider option ε =0.1, H=1, L=4, H₁=0.05, H₂=0.238 for which Q=42, H₀=0.75 is received, and, therefore, relative errors make respectively 71 and 61%..

Thus, as well as in 9, here too evaporation significantly influences a current picture.