On Filtration in a Rectangular Interchange with a Particularly Unpermatable Vertical Wall in the Evaporation

We consider a plane steady-state filtration in a rectangular bridge with a partially impermeable vertical wall in the presence of evaporation from a free surface of groundwater. To study the effect of evaporation, a mixed multiparametric boundary-value problem of the theory of analytic functions is formulated and using the method of P. Y. Polubarinova-Kochina. Based on the proposed model, an algorithm is developed to calculate the dependence of efficiency and productivity of hydrodynamic analysis.


Introduction
As it is known [1][2][3][4][5][6] , the exact solution of tasks on inflow of liquid to an imperfect well with the flooded filter (i.e. an axisymmetric task) or the tubular well representing an impenetrable pipe with the filter in its some part is connected with great mathematical difficulties and so far isn't found. Therefore in due time as the first approach to the solution of similar tasks some corresponding flat tasks analogs about a filtration to imperfect rectilinear gallery in free-flow layer [4,7] and in a rectangular crossing point with partially impenetrable vertical wall were considered [8] . It should be noted that areas of complex speed of the specified cases allow to apply by means of inversion at the decision Christoffel-Schwartz's formula.
In work [9] it is shown that the current picture near the impenetrable screen significantly depends not only on imperfection of gallery, but also on evaporation existence that is strongly reflected in an expense of gallery and ordinate of a point of an exit of a curve depression to an impenetrable wall.
In the real work the exact analytical solution of a task on a current of ground waters through a rectangular crossing point with partially impenetrable vertical wall in the presence of evaporation from a free surface of ground waters is given. In this case in the field of complex speed, unlike [1,4,[6][7][8] there are not rectilinear, but circular polygons that doesn't give the chance to use classical integral of Christoffel-Schwartz.
The accounting of characteristics of the considered current allows to receive the decision through elementary functions that does its use by the simply and convenient. The provided detailed hydrodynamic analysis gives the flavor about possible dependence of filtrational characteristics of the movement on all physical parameters. The received results, at least, qualitatively can be postponed for a case of tubular wells.

Formulation of the problem
2 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.
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: AD 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.
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 πv = 2arctg ε 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, [9,21,22] ).
The function making conformal display of a semi-strip to area of complex speed of w, has a former appearance [9] ε (ch ch sh sh ) (ch sh sh ch ) ε , ch ch sh sh ε (ch sh sh ch ) where С (C  1) -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] , 3 considering that w = dω / dz and in view of a ratio (2), we will come to dependences and also required coordinates of points of a free surface AD 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.

Calculation of the scheme of a current and analysis of numerical results
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 of Q ordinate of H0 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).
Results of calculations of influence of the defining physical parameters ε, H, H1, H2 and L at sizes Q and H0 are given in  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. 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.
Thus, as well as in [9] , here too evaporation significantly influences a current picture.

Conclusion
The technique of creation of the exact analytical solution of a task on the movement in liquid in a rectangular crossing point with the screen in the presence of evaporation from a free surface of ground waters is developed. It is shown that the current picture near the impenetrable screen significantly depends not only on the filter size, but also on evaporation existence that is strongly reflected in an expense and ordinate of a point of an exit of a curve depression to the screen. The received results give an idea (at least qualitatively) of possible dependence of characteristics of a current by consideration of a task about a filtration already to an imperfect well or a tubular well.