Modeling the Movement of Groundwater from the Pits, Surrounded with Tongues of Zhukovsky

In the hydrodynamic statement the filtration low from ditches, walled tongues of Zhukovsky is considered. The fluid moves through the layer of soil, underlain by a well-permeable pressure aquifer, which is contained waterproof area on the roof. For study the infiltration to the free surface of groundwater is formulated a mixed multi-parameter boundary value problem of the theory of analytic function, which is solved by the Polubarinova-Kochina's method and ways the conformal mapping areas of a special kind, which are characteristic for tasks of an underground hydromechanics.


Introduction
In the hydrodynamic formulation is considered flat established incompressible fluid filtration by Darcy's law in construction ditches fences tongue Zhukovsky through homogeneous and isotropic soil layer underlained by a wellpermeable pressure aquifer on the roof that provides an impermeable land. During the study infiltration of the free surface groundwater formulated mixed boundary multiparameter problem of analytic function theory, which is solved by the method Polubarinova-Kochina and methods of conformal mappings of a special type, typical of underground fluid mechanics problems. Based on this model, an algorithm of calculation the filtration characteristics in situations when you have to take into account the combined effect of the picture movement of such important factors as the infiltration of the free surface, tight inclusion and backwater from the water well-permeable underlying aquifer. Using the exact analytical dependences and numerical calculations carried out hydrodynamic analysis of the structure and features of the modeled process and the effect of all physical parameters of the circuit on the filtration characteristics. The limiting case of flow associated with the absence of a backwater opaque area or infiltration and degeneration of ditches in a semi-infinite strip on the left of flooding. We give a solution of the problem for the circuit assuming a finite value of flow velocity at the tip of the tongue, which is an analogue of the classical problem of Zhukovsky. The results of calculations for all limiting cases are compared with the main filter model.
The study of filtration flows from the construction pits, fenced symmetrical tongue Zhukovsky, related to work [1][2][3][4][5][6][7][8][9][10][11]. It was assumed that the water-permeable layer of soil has unlimited power in some cases, in others underlying wellpermeable pressure reservoir was modeled by one or two drains in the form of a horizontal slit Zhukovsky [17]. In some studies examined free filtration, that is, for no backwater, and in some cases -the pressure, the presence of the free surface of neglect. In all these studies, infiltration records are not made. There were used function of Zhukovsky and method of Vedernikov-Pavlovsky, which reduce the case to a conformal mapping of rectilinear polygons and then using the Schwarz-Christoffel formula.
As shown by the practical application of these methods [12][13][14][15] their direct use only lead to effective results when the boundary of the movement consists of horizontal and vertical watertight permeable areas. However, in actual hydraulic construction pits (canals, reservoirs) immediately below the overburden, together with the horizontal aquifers higher permeability (pebbles, gravel, coarse sand) often occur also waterproof horizontal sections (tight turn, impermeable rock), that fundamentally affects the nature of the filtration processes. In such situations, the use of the Schwarz-Christoffel integral does not lead to the goal, as in the areas of integrated flow rate already arise circular polygons, cannot be reduced directly to the straight line.
In contrast to these researches below, as a direct continuation and development of the author's previous results [18][19], we study the problem of the flow of fluid from the pits through the dirt array ultimate power, underlain by a well-permeable pressure aquifer that comprising at its waterproof roof portion of infiltration in the presence of the free surface. We study the most general case of motion in which both permeable areas of the boundary flow filtration takes extreme values and point of zero flow rate enters the tongue. (that apparently has not yet occurred in the literature). There have been extreme cases of course, related to the lack of a backwater, a tight turn or infiltration, and the case of degeneration foundation ditches in a semi-infinite strip on the left of flooding previously studied by V.V. Vedernikov [10]. The results for the circuit that occurs in the absence of critical points in the case where the flow rate at the end of the tongue is finite; the resulting solution is an analogue of the classical problem of Zhukovsky [17].
For the solution of a mixed boundary value multiparameter problem of theory of analytic functions used method Polubarinova-Kochina [12][13][14][15][16] and areas designed for specific species [20][21] of conformal mapping of circular polygons [22][23][24], which are typical for problems of underground hydromechanics. Accounting for the characteristics of movement allows us to represent the solution through a special, and in some cases, elementary functions, making them easy and convenient use.
Based on the constructed exact analytical dependences and numerical calculations performed by the hydrodynamic analysis of the influence of the physical parameters of scheme on picture effects and highlights some of the features developed models. Mathematical modeling results for all extreme cases compared with the primary filter circuit. Highlights were announced earlier [25].

The Basic Model: Formulation of the Problem
We consider the planar steady flow from the excavation A'A width 2l, fenced symmetrical tongue Zhukovsky same length S through the permeable capacity of soil layer T with the underlying aquifer E'E, containing or underground artesian water wherein the pressure has a constant value H 0 ( Figure 1). On the roof of the reservoir is waterproof section D'D, simulated horizontal segment of length 2L. By the symmetry of pictures motion, we shall restrict the right half of the study filtration area ABCDEGR. Ground water, flowing tongue ARG under the influence of the difference of pressures in the trench and the underlying good permeable aquifer, raised him to a certain height RG and breaking point M zero speed on the tongue, form the free surface of GE, which receives infiltration water intensity ε (0 <ε <1), referred to the soil filtration coefficient κ = const. The task is to determine the position of the curve GE depression, and, therefore, raise the height of the RG groundwater of the tongue, that is, the value of S -d.
We assume that the movement of groundwater obeys the Darcy's law with a known filtration coefficient κ and takes place in a homogeneous and isotropic soil, which is considered incompressible. The rate of flow at the tip of the tongue v R assumed infinite (up to s. 5), the depth of water in the trench H remains unchanged over time.
From a mathematical point of view the problem is to find the complex potential flow ω = φ + iψ, (φ -potential speed, ψ -current function) as an analytic function in the field of filtration of the complex coordinates z = x + iy with the following boundary conditions: where Q -the desired filtration rate from the pit. A study carried out in terms of the above variables w and z, associated with the same name and the actual values ω f and z f by

Construction of the Solution of Boundary Value Problem
Turning to the field of complex velocity w ( Figure 2), corresponding to the boundary conditions (1). This region, which is circular polygon with three cuts, the vertices N 1 and N 2 of two of which correspond to the extremes of the stream function at impermeable portions AB and DE, belongs to the class of polygons in polar grids [26] and coincides with that in the case previously examined [19] (Figure 2). However, in contrast to [19], in this case, on the border of the movement area, an additional corner point -the point B; the total number of singular points is equal to nine, which greatly complicates the task.
To solve the boundary value problem we use the method of Polubarinova-Kochina, which is based on the application of the analytical theory of linear differential equations of the Fuchs class [12,16,27]. We introduce the auxiliary parametric variable ζ and function z(ζ), conformally maps the upper half in the region z accordance with points Defining characteristic indicators of functions Ω and Z near the regular singular points [12], we find that they are linear combinations of the two branches next Riemann function [12,27]: It is evident that the point ζ = ζ A and ζ = ζ B -ordinary point of function Y, which represents the last symbol of Riemann. This symbol of Riemann corresponds to a linear differential equation of Fuchs class with seven regular singular points, which Surrounded with Tongues of Zhukovsky is quite typical for the problems of underground hydromechanics [20,21] and in this case has the form 1 2 Recall that in addition to affix ζ F , ζ N1 and ζ N2 in equation (4) accessory parameters λ 0 , λ 1 and λ 2 are unknown at the statement of the problem and should be identified during its solution.
Taking into account the relations (3), (5) and (6), as well as the fact that the function w = dω / dz has previous form [18,25] we arrive at the source dependencies Here N > 0 -constant scale simulation, A = sn(2Ka, k'), B = sn(2Kb, k'), a and b -unknown ordinates of points A and B domain τ. The representations (9) constant conformal mapping α, β and γ, which are connected by the relation (8), subject to the conditions 0< α < r < β < m < a < b < ρ/2, 0 < γ < ρ/2, (10) regulating the position in the current boundary field points of zero velocity M and the tip of the tongue R, and the well N 1 and N 2 ; m and r -unknown ordinates of the points M and R in the plane τ.
You can verify that the functions (9) satisfy the conditions (1), reformulated in terms of functions dω/dτ and dz/dτ, and thus, are parametric solution of the original boundary value problem.
Writing equations (9) for different parts of the border region τ followed by integration over the whole contour of the sub-area leads to the closure motion field and thus serves as a computation control.
The result is an expression for defined and the desired geometrical and filtration properties of the model and coordinate EG depression curve points: In the formulas (11) - (14) integrands -expression right side of (9) in the respective sections of the circuit area T.

The Numerical Results for the Main Filter Model: Discussion of Results
The representations (9) The first of these relations means that the rate at the end of the tongue tends to infinity, and the second follows directly from consideration of the boundary conditions (1). After determining the unknown constants are unknown quantities d and Q from the formulas (12) and, finally, by the formulas (13) calculated coordinates of the free surface EG points.   First of all, attention is drawn to the same qualitative character of dependent variables d and Q on the parameters T and l, S and H, and at the same time the opposite of the behavior desired characteristics when changing the parameters S and H on one side and on the other -L and H 0 .
Increased infiltration intensity, width tight turn and head in the underlying reservoir and power reduction layer of the tongue length, water pressure in the pit and its width lead to reduction of the depth d, which increase ordinate of point G yield curve depression from under the tongue.
Thus, according to the Table 1 and 2, an increase in the parameters ε, S, H 0 and l in the 1.6, 2.0, 2.0 and 1.1 times accompanies the change in the value of d in 2.2, 1.1, 1.3 and 1.6 times, respectively. However, the greatest impact on the depth d has an impermeable area: Data Tables 2 show that increasing the width L of 28% total depth d increases almost 10-fold.  As for consumption, the tight turn with increasing width of the Q-value is also increasing: Table 2 shows that an increase in the parameter L to 1.28 times the flow rate implies an increase of more than 12 times. Thus, it revealed a significant propping up effect of impervious area in relation to the filtering of the pit.
This flow behavior is clearly observed how with increasing layer thickness, the width of the pit, and the pressure in the underlying horizon, and with a decrease in the length of the tongue and head to tail water. Table 2 shows that reduction of the H only 2 times accompanied by an increase in flow rate Q is almost 48 times, indicating that the greatest impact on the consumption of water pressure in the pit. incision EG field w, transforming, degenerates in the right semicircle (dashed line in Figure 2) and thus the source area is transformed into a circular hexagon, which falls right side of the semicircle |w -i(1 + ε)/2| < (1 -ε)/2. In the course of the plane z for * γ γ = depression curve flattens out at point E, deviates from that merges its inflection point F and exits to the roof of the underlying horizon at right angle. Figure 4 shows the flow pattern is calculated with the basic version ε = 0.6, T = 7, S = 3, L = 15, H = 5, l = 10. Table 3    A comparison with the results for the base model shows that there is a completely different behavior of the values of d and Q by varying the parameters of S and H: now with increasing length of the dowel head and tail-to the contrary, d depth decreases and the flow rate Q increases. Moreover, the behavior of the quantities Q and d, parameters ε and L becomes qualitatively similar, and the parameters ε and lopposite.

Limiting Cases
Just as in the main filter circuit, a considerable influence on the value d has a water pressure in pond: Table 4 shows that change the argument H on 50% corresponds to a decrease of depth d almost 8 times. The greatest impact, as before, has a tight inclusion. Table 4 also shows that change the width L of just 1.3 times the flow rate implies a reduction by more than 18 times.  flow results obtained previously [18,25]. 3 0 . Case L = 0. In the main filter scheme discussed above, the absence of such a powerful factors as the tight incorporation, being limiting, serves as a background in the evaluation of the role of infiltration and pressure in the underlying reservoir and allows more contrast to trace the interaction of important physical parameters, such as ε and H 0 .
If there is no impervious area on the roof of the underlying aquifer, the latter along its entire length becomes wellpermeable. When you merge points C, D in the area of complex velocity her left half-plane w cut off, a circular incision EG moves in the right half-plane, and source area is transformed into a circular triangle ( Figure 5). In the movement plane z point D, merging with point C, yield on the axis, and rectangle of the plane τ is transformed to halfstrip 0 < Reτ <1/2, 0 < Imτ < ∞, because parameter ρ =K′/K = ∞, K = π/2. The solution for this limiting case is obtained from the formulas (9)- (14), if we put k=0 in these formulas and consider that in this case the elliptic functions degenerate into trigonometric and theta-functions are cut off in their first terms or constants: Here arcsin m m ′ = , arcsin r r ′ = , m′, r′, c′ (0 < m′ < r′ < a′ < b′ < c′ < π/2) are inverse images of points M, R, C on the horizontal axis of the plane τ. This case is described in detail previously [19], where an analysis of the influence all of the physical parameters of the model. 4 0 . Case ε = 0. Within the case L = 0 we focus on the absence of infiltration. Given the connection between the parameters m' and r', noted in (16), we see that in the case where ε = 0, there is implying the solution of the problem of dependences (16) where m′ = 0 that is when the plane τ point depression curve G, E are merged at the origin with point M zero speed. There is a solution to the problem, first examined V. V. Vedernikov [10], only in the other way.

The Case of a Finite Value of Flow Rate at the End of the Dowel: Analysis of Zhukovsky Problem
Within the boundary value problem (1) we consider the case when the flow speed on the end of a tongue v R , 0 < v R < ε finite and stream function in the permeable sections AB and DE has no extremes. Then, in the complex velocity w there are disappearing both of the vertical incisions, the left halfplane is truncated, as previously in the case of L = 0, but, unlike the latter, MR portion is transferred to the first quadrant (dashed line in Figure 5). As a result, the source region is transformed into a circular pentagon. Parametric solution of the problem is formally the same form (9) with the replacement of the integrals Y 1,2 (τ) and regular conformal mapping α and β on next [29,30]: (1 ) 2 i A similar solution to the problem in the case of lack of backwater flows from ideas (9), (17), (18) The analysis of the numerical results shows that in the case of v R < ∞ retained the qualitative nature of the dependencies of the filtration rate of the physical parameters of the circuit typical case when v R < ∞. For example, there is the same as before, the flow behavior of T and l values from one side and the opposite character of S and H parameters -on the other. Significant impact on consumption on Q, and as before, have infiltration, a dense layer switch and power. Figure 6 shows the pattern of motion calculated at ε = 0.5, T = 6, S = 3, H 0 =0, L=16.2, H = 3, l = 15.
Noteworthy is the fact that all settlement options is d = S, and therefore, the value of h(d) = h(S) = 0. This means that in the plane of the current point G yield curve depression out of the tongue merges with the R point of his sharp; from the review of the field, comprehensive rate w implies that in this case the speed at the end of the tongue is equal to the infiltration rate: v R = ε, 0 < ε < 1.
If you make the transformation τ′ = 1/2 + iρ′τ, sending rectangle auxiliary variable τ in the like with parameter ρ′ = 1/ρ = K/K′, then the corresponding primary filter circuit on the parameters of inequality (10) takes the form: where b′, a′, r′ -abscissa's inverse images of points B, A, R in the plane τ. Calculations show that for any value of the intensity of infiltration ε (0 < ε < 1) the ratio of d = S holds only for single values of r′ -its limit * r′ , when the plane τ' merge point G, and R: All other valid values lead to inconsistencies with the real picture of the flow -the relationship d > S, i.e., the separation of the flow. A similar result in the limit for this model when the water permeable layer of soil has unlimited power, there is no impenetrable plot and infiltration, when T = ∞ (k′ = 0, k = 1), L = 0 ( * 0 b b ′ ′ = = ) и ε = 0 (m′ = 0), was first obtained in due time, N. E. Zhukovsky [17]. The solution for this limiting case is obtained from dependencies (9), (17), (18), if you put in them K = ∞, K′ = π/2, k′ = 0, k = 1, b′ = 0, q′ = 0 and consider that in this case the elliptic functions degenerate into hyperbolic, and theta-functions which, this time characterized by the parameter q '= 0, break off on their first terms or constants. Thus, in the limiting case study scheme Zhukovskogo obtained solution of the problem only by other means.

Conclusion
Executed in consideration of flows of pits transformed from the basic filter circuits may serve to illustrate the variety of physical content multiparametric boundary value problem with a free surface. An important place is occupied with the extreme cases that seem to be bordered by the original simulated process in describing its boundary value problem and lead to transformations considered the main filter circuit. Access to such extreme cases is carried out on reaching any of the unknown parameters of a conformal mapping of its critical values.