On Filtration in a Rectangular Interchange with a particularly Unpermatable Vertical wall in the Evaporation

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

Figure 1.The current picture calculated at ε=0.5, H=3, L=3, H1=1.0, H2=1.4.

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 ^{1, 2, 3, 4, 5, 6, 20}. 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 ^{12, 13, 14, 15, 16, 17, 18, 19} 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, ^{9, 21}).

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 ^{1, 2, 3, 4, 5, 6, 20}, 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.

Figure 2.Dependences of the sizes Q and H0 from ε (а) at H = 3, L = 2, H1 = 1 H2 = 1.4, , from H (б) at ε = 0.5, L = 2, H1 = 1, H2 = 1.4; от L (в) at ε = 0.5, H = 3, H1 = 1, H2 = 1.4; from H1 (г) at ε = 0.5, H = 3, L = 2, H2 = 1.4; from H2 (д) при ε = 0.5, H = 3, L = 2, H1 = 1.

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.