On ground water seepage under hydraulic structures

ISSN 2070�0482, Mathematical Models and Computer Simulations, 2011, Vol. 3, No. 5, pp. 619–628. © Pleiades Publishing, Ltd., 2011.Original Russian Text © E.N. Bereslavskii, L.A. Aleksandrova, E.V. Pesterev, 2011, published in Matematicheskoe Modelirovanie, 2011, Vol. 23, No. 2, pp. 27–40.

619

1. INTRODUCTION

Paper [1] was concerned with two mathematical models of flows under a subsurface dam andJoukowski’s cutoff wall. The first problem dealt with simulation of a smooth underground contour of ahydraulic structure aimed at determining the form of an underlying water pervious abutment of a curvi�linear blanket of constant incident velocity. In the present paper, we focus on the case when a blanket withsimilar features has a more involved configuration and consists of one horizontal and two curvilinear ele�ments. It is worth noting that introducing such curvilinear elements makes it possible to avoid consideringnonphysical semi�infinite and infinite domains; this is of special value in developing approximate andnumerical methods (the finite elements method, the boundary integral equation method, etc.). The sec�ond problem of paper [1] was concerned with the flow past a Joukowski cutoff wall through a watered soillayer into an underlying highly permeable horizontal aquifer devoid of confined groundwater (seepagewithout backwater or so�called free seepage), the left semi�infinite portion of whose roof was simulated byimpermeable inclusion (hard rock, blanket). In the present paper we consider the case, most often met inapplications, when the underlying highly permeable formation contains confined groundwater (seepagewith backwater).

Multiparameter problems of the theory of analytic functions are formulated in order to study thesemodels; solutions are obtained using conformal mappings of domains of a special form. A constructivesolution of the problems in question is obtained and explicit analytic representations for typical dimen�sions of the flow patterns are ascertained. The results of numerical calculations are presented and thehydrodynamic effect of the physical parameters of models on the dimensions of flow domains is deter�mined. The limiting cases (previously examined in [1]) when there is no horizontal portion of the blanket(in the first diagram) and no backwater in the underlying highly permeable formation (in the second dia�gram) are singled out.

On Ground Water Seepage under Hydraulic StructuresE. N. Bereslavskii, L. A. Aleksandrova, and E. V. Pesterev

St. Petersburg State University of Civil Aviation, ul. Pilotov 38, St. Petersburg, 196210 Russiae�mail: [email protected], [email protected], [email protected]

Received May 27, 2010

Abstract—This present paper continues paper [1] of the authors and is concerned with the followingproblems regarding seepage flows with unknown boundaries under hydraulic structures, with use beingmade of the theory of two�dimensional (2D) steady state seepage of incompressible fluid according toDarcy’s law in a uniform isotropic soil. Below we construct a smooth underground contour of a rect�angular subsurface dam of constant seepage rate in the case when a pervious abutment is underlain bya blanket consisting of two curvilinear and one (middle) horizontal elements which are also character�ized by constant incident velocity. We are concerned with the fluid flow below Joukowski’s cutoff wallthrough the watered soil mass underlain by a highly permeable aquifer with confined groundwater; theleft semi�infinite part of its roof is simulated by impermeable inclusion. These motions are examinedusing multiparameter problems of the theory of analytic functions. Their solutions are obtained by thesemi�inverse method of the velocity hodograph by P.Ya. Polubarinova–Kochina and I.N. Kochina, byPolubarinova–Kochina’s approach dependent on the analytic theory of linear Fuchsian equations,and using the available conformal mappings of domains of special kinds peculiar for undergroundhydrodynamics. The results of the numerical calculations are given and the hydrodynamic effect of thephysical parameters of models on the flow pattern is ascertained. The limiting cases previously exam�ined in [1] are singled out.

Keywords: seepage, ground waters, dam, cutoff wall, complex velocity region, conformal mapping.

DOI: 10.1134/S2070048211050061

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BERESLAVSKII et al.

2. SIMULATING THE UNDERGROUND CONTOUR OF A HYDRAULIC STRUCTURE FEAUTURING PORTIONS OF CONSTANT INCIDENT VELOCITY

We consider two�dimensional (2D) steady state seepage of incompressible fluid (according to Darcy’slaw with the given seepage coefficient κ = const) in uniform isotropic soil under an impermeable under�ground contour of a subsurface dam ABCC1B1A1 (Fig. 1). The region of the motion is bounded from belowby the blanket G1G consisting of two curvilinear elements G1F1 and GF and (as distinct from the case con�sidered in [1]) of a horizontal element F1EF; the incident velocity is assumed constant on these elementsas well as on the other elements of the underground contour of the hydraulic structure BC and B1C1.

If we introduce the complex�valued motion potential ω = ϕ + iψ (Fig. 2) and the complex coordinatez = x + iy in relation to κH and H, respectively, where H is the pressure head exerted on the structure, thenthe problem in question amounts to determining the position of the curves BC, B1C1, G1F1, and GF underthe boundary�value conditions

(1)

in order that the seepage rate along the curvilinear elements of the underground dam contour BC andB1C1, as well as along the horizontal FEF1 and curvilinear elements G1F1 and GF of the blanket, have con�stant values v0 (given) and u0 (sought), respectively (0 ≤ u0 < v0).

Consider the complex velocity w depicted in Fig. 3a. This is a domain with right angles and cuts, whichcan be looked upon as a polygon in polar grids [4], and which differs from that of [1] in having a horizontalcut F1EF along the real semiaxis of the w plane; this permits us to use the Riemann–Schwarz principle of

A1G1: y 0, ϕ 0.5H, A1B1: x– l, ψ– Q,= = = =

C1DC: y d– , ψ Q, AB: x l, ψ Q,= = = =

AG: y 0, ϕ 0.5H, F1EF: y T, ψ– 0,= = = =

B1C1, BC: w v0, G1F1, FG: w u0, ψ 0,= = =

y

z

l

Ll2

A1G1 A Gxd B1

C1l1 l1

D C

B d1

2

1

21

l

T

F1E

l3 l3 F

Fig. 1. The flow pattern as calculated for v0 = 1, H = 2, Q = 1.14, T = 1.934, Δl = 0.308, and Δd = 0.295.

1

0.5


ON GROUND WATER SEEPAGE 621

symmetry in order to substantially eliminate unknown constants of the conformal mapping. Hence, tak�ing rectangle [5] (Fig. 3b) for a canonical domain and taking into account the complete symmetry in thez, ω, and w domains, we can confine ourselves to considering the region of the motion ABCDEF and thecorresponding domains in the ω and w planes. Hence, since the domain of complex velocity w = dω/dzcoincides with that of [1], we have

(2)

which gives the physical parameter u0 = v0 exp(–0.5πρ).

The conformal mapping of the rectangle with respect to the auxiliary variable τ into the complexpotential domain ω is given by

(3)

w τ( ) v0 τ 0.5–( )πi,exp=

ω 0.5K k( )��F λ

n�� 1 n2

sn 2Kτ k,( )2–

1 λ2sn 2Kτ k,( )2–

�� m,arcsin .=

(a) (b)

B1

A1

G1

0

G

A

B

w

F1E D

F

|w| = v0

v

G

0.5 ρ A

B0

ac

b

F

E

D

C

0.5 u

|w| = u0

τ

C1

C

Fig. 3. The domain of auxiliary parametric variable τ(a) and of the complex velocity w(b).

A1

G1

D A

Qω

0 E G ϕ0.5 H

ψ

Fig. 2. The complex potential domain ω of the flow.

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BERESLAVSKII et al.

In this formula, F(ϕ, m) is the elliptic integral of the first kind of the modulus m = kn[(1 – k'2α2β2)/(1 –k'2α2γ2)]1/2/λ, λ = [1 – k'2β2]1/2, n = [1 – k'2γ2]1/2, α = sn(2Ka, k'), β = sn(2Kb, k'), and γ = sn(2Kc, k').Also, the following condition must be satisfied

(4)

this condition bounds the physical parameters Q and H and serves to determine modulus k.Taking into account relations (2) and (3) and repeating the arguments of [1], we obtain

(5)where M is the scaling constant of modeling (M > 0). Writing representation (5) for different parts of theboundary of the domain τ and integrating over the entire contour of the auxiliary domain, we obtain thefollowing expressions for basic geometrical and seepage characteristics

(6)

for the coordinates of the points of the underground contour of the hydraulic structure, BC, we have

(7)

and for the coordinates of the points of the curvilinear part of the blanket FG, we have

(8)

Here, Δl = l – l1, Δd = d – d1, XBC, YBC, ΦEF, ΦFG, XFG, and YFG are the expressions for the right�hand sidesof (5) on the corresponding parts of the contour of the τ plane.

Putting t = 0.5 in Eqs. (6) and (7), we obtain the sought dimensions of the underground contour of thedam and of the curvilinear blanket:

(9)

In the direct physical formulation, the parameters of the conformal mapping α, β, γ, and M are theones sought; they can be determined via Δl, Δd, H, and T given in terms of Eq. (6). Numerically, one canshow that the functions appearing in the left�hand sides of these equations are monotonic; therefore, theseequations are uniquely solvable in terms of the constants sought. In doing so, we preliminarily eliminatethe modeling constant M from all of the equations by using the third equations of system (6), which fixesthe quantity H = 1.

The limiting cases. We first dwell upon the case when the points F1, E, and F merge into one in the planeof flow; i.e., when the horizontal impermeable part is not present and so the blanket is curvilinear through�out [1]. In this case, the parameter γ in the τ plane is γ = 0.5ρ, and now the solution results from formulas(4)–(9), if we take γ = 1.

The other limiting case is when the blanket is horizontal [2, 3] as far as it extends. Then, in the z planeof motion, points G and F, as well as G1 and F1, merge into one at infinity, with the rectangle in the τ planedegenerating into a half�strip. The solution in this case results from formulas (4)–(9), on taking k = 0; theexpressions for H and T can be integrated explicitly

(10)

These formulas coincide, with accuracy up to notations, with the available ones (see [3], formulas(7.17) and (7.18)).

ρ m( ) K ' m( )K m( )�� 2Q

H��,= =

dωdτ�� Mf τ( )

Δτ��, dz

dτ��– Mf τ( )e 0.5 τ–( )π i

v0Δ τ( )��,–= =

f τ( ) sn 2Kτ k,( )cn 2Kτ k,( )dn 2Kτ k,( ),=

Δ τ( ) 1 λ2sn 2Kτ k,( )2–[ ] 1 n2

sn 2Kτ k,( )2–[ ] α2 1 α2–( )sn 2Kτ k,( )2+[ ],=

Mv0

�� XBC td

0

0.5

∫ Δl, Mv0

�� YBC td

0

0.5

∫ Δd, M ΦEF td

0

0.5ρ

∫ ΦFG td

0

0.5

∫+⎝ ⎠⎜ ⎟⎛ ⎞

0.5H, Mu0

�� YFG td

0

0.5

∫ T,= = = =

xBC t( ) l Mv0

�� XBC t, yBC t( ) d1– Mv0

�� YBC t, 0 t 0.5,≤ ≤d

0

t

∫–=d

0

t

∫–=

xFG t( ) L Mu0

�� XFG t, yFG t( ) Mu0

�� YFG t, 0 t 0.5.≤ ≤d

0

t

∫–=d

0

t

∫–=

l1 xBC 0.5( ), d1 yBC 0.5( ), l2Mv0

�� ΦFGeπ t t, l3d

0

0.5ρ

∫ L xFG 0.5( ).–= = = =

H 2MK k( )

π 1 α2β2–( ) 1 γ2–( )��, T M

v0 1 α2–( ) 1 β2–( ) 1 γ2–( )��, k= 1 α2β2–( ) 1 γ2–( )

1 α2γ2–( ) 1 β2–( )��.= =



Figure 1 depicts the flow pattern as calculated with the following base values: v0 = 1, H = 2, Q = 1.14,T =1.934, Δl = 0.308, and Δd = 0.295. The calculations of the effect of the generating physical parametersv0, H, Q, T, Δl, and Δd on the dimensions l1, d1 (and hence on l and d), l2, and l3 are given in Tables 1–3.In each of the blocks of the tables, one of the parameters in question ranges over the admissible values, andthe remaining ones are equal to the base values. The following conclusions can be made from Tables 1–3.

A decrease in the incident velocity and increase in the pressure head exerted on the structure increaseall the dimensions of the dam as well as the lengths of the horizontal portion of the blanket. From Table 1it follows that if the velocity changes to 1.4 times, the width l1 and thickness d1 increase to 133 and 218%,respectively; the pressure head has the most pronounced effect on the width and depth of the hydraulicstructure: it is seen from the second section of Table 1 that if parameter H is increased to 50%, the quan�tities l1 and d1 increase by factors of 4.6 and 3, respectively.

From the first section of Table 2, it follows that the seepage flow rate has practically no effect on thedam’s dimensions. At the same time, there tends to be an increase in the width l1 of the structure as theseepage flow rate Q increases and the formation depth T decreases (second section of Table. 2); on theother hand, the depth d1 increases as the parameter Q decreases and T increases. It is worth pointing outthat alongside with the parameter H, the depth of the formation also has a profound effect on depth d1,changing the latter by a factor of 4.6.

The sections of Table 3 relating to parameters Δl and Δd reflect the following tendency: an increase inthe difference Δl (Δd, respectively) is associated to a decrease (increase) of the width l of the dam and anincrease (decrease) of its depth d. This being so, if Δl changes to 47%, the width l1 declines by a factor of3.4 and the depth d1 increases 2.3�fold, and if Δd changes to 40%, the width l1 increases by 2.3�fold, whilethe depth d1 decreases by a factor of 8.8. The last row of Table 3 corresponds to the flow due to the cutoffwall (tooth), when l1 = 0, l = Δl, and the dam’s apron with a horizontal insertion, where d1 = 0 and d = Δd(see [3]).

Of special importance is the water uplift behavior in the downstream wall l2 and the dimensions of thehorizontal portion of the blanket l3. As is seen from Table 2 and Fig. 4, II, the width l2 increases as Q andT increase; on the other hand, l2 decreases as v0, H, Δd, and Δl increase. Here, the values of l2 and l3 canbe quite considerable: for Q = 2, we have l2/l1 = 3.9, l3/l =3.6, l2/d = 7.3, and l3/d = 6.6.

Table 1.

v0 l1 d1 l2 l3 H0 l1 d1 l2 l3

0.850 0.838 0.375 2.038 1.835 0.12 0.085 0.076 2.646 0.900

0.100 0.397 0.161 1.640 1.051 0.16 0.273 0.186 2.228 1.160

0.120 0.359 0.118 1.026 1.018 0.18 0.395 0.230 2.072 1.288

Table 2.

Q l1 d1 l2 l3 T l1 d1 l2 l3

1.4 0.571 0.226 2.426 2.014 1.1 0.710 0.055 1.265 1.543

1.8 0.597 0.200 3.192 2.846 1.7 0.598 0.202 1.795 1.626

2.0 0.601 0.195 3.579 3.245 1.9 0.553 0.246 2.020 1.668

Table 3.

Δl l1 d1 l2 l3 Δd l1 d1 l2 l3

0.30 0.298 0.290 2.233 1.509 0.25 0.340 0.588 2.135 1.519

0.44 0.088 0.676 2.174 1.331 0.35 0.721 0.066 1.949 1.600

0.50 0 0.735 2.173 1.254 0.40 0.757 0 1.928 1.618

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BERESLAVSKII et al.

3. SIMULATING THE FLOW DUE TO THE JOUKOWSKI CUTOFF WALL THROUGH

A WATERED SOIL LAYER UNDERLAIN BY A HIGHLY PERMEABLE CONFINED AQUIFER

The case in question is depicted in Fig. 5. We are con�cerned with the fluid flowing under the Joukowski cutoffwall through a watered (with uniform seepage intensityε, 0 < ε <1) soil layer of depth T into an underlying highlypermeable aquifer of constant pressure head H0; the leftsemi�infinite part BC of the formation roof is modeledby an impermeable inclusion (a blanket, hard rock, etc.).The cutoff wall AGF is in a stream of ground waterexerted to the pressure head differential at the upstreamwall and the lower highly permeable soil layer; down�stream of the cutoff wall, the water uplifts to some heightGF to form a free surface DF. However, as distinct from[1], here the flow pattern changes substantially: the pres�ence of backwater from the underlying aquifer intro�duces an additional boundary point in the seepage pat�

tern—this is flex point E of the free surface DF. The fact introduces a serious complication in solving thecorresponding boundary value problem, increasing the total number of unknown parameters of the con�formal mapping. In the case in question, the problem consists in finding the position of the seepage lineDF under the following boundary conditions:

(11)

here, Q is the seepage flow rate. Finding the uplift height downstream from cutoff wall GF (quantity d), aswell as the position of the abscissa of point C (quantity L), has some practical utility. The available pressurehead H, the depth of the soil layer T, the length of the cutoff wall S, and the incident velocity at its end VG

(0 < VG < ε), as well as the parameters H0 and ε, are considered as given.

The domain of complex velocity w, which corresponds to the boundary conditions (11) and which isdepicted in Fig. 5a, has right angles and a circular cut; hence, it belongs to the class of circular–arc poly�gons in polar meshes (see [4]). Hence, taking again the rectangle in the τ plane for an auxiliary parametric

AB: y 0 ϕ, H, BC: y– T– ψ, 0, CD: y T– ϕ, 0,= = = = = =

AFG: x 0 ψ, Q, DEF: ϕ y– H0 T, ψ–+ εx Q,+= = = =

y

xBH

A 0

dF

S

T

B L C H0 D

Fig. 4. The flow pattern under the cutoff wall.

Fig. 5. The domain of complex velocity w(a) and of the auxiliary parametric variable τ(b).

G

E

(a) (b)v

C

D1F

εA

B C u

w

F

G

0.5ρ

A

B

a

gτ

0.5

C

D

E



variable (Fig. 5b) and applying the approach elaborated in [5–8] for constructing the mapping functionsfor such polygons, it is found that

(12)

here, ϑ2(τ) is the second theta function with parameter q = exp(–πρ), which is a single�valued functionof modulus k (see [9]).

Using the method of P.Ya. Polubarinova–Kochina [3], which is based on the analytic theory of lineardifferential equations [10] and taking into account the relations w = dω/dz and (12), the solution of theboundary value problem (11) can be written in the following parametric form:

(13)

Here, α1 = (1 – α2)1/2, α = sn(2Ka, τ), and a is the ordinate of the point A in the τ domain.

In the case in question, the unknown constants of conformal mapping α (or a), ordinate g of point Gin the τ domain, modulus k, and the scaling constant of modeling M are determined by solving the follow�ing system of equations:

(14)

w τ( ) εiϑ2 τ iλ+( ) ϑ2 τ iλ–( )–ϑ2 τ iλ+( ) ϑ2 τ iλ–( )+��, λ arth ε

π��,= =

dωdτ�� i εM

ϑ2 τ iλ+( ) ϑ2 τ iλ–( )–ϑ1 τ( )dn 2Kτ k,( )Δτ

��, dzdτ�� M

ϑ2 τ iλ+( ) ϑ2 τ iλ–( )+ϑ1 τ( )dn 2Kτ k,( )Δτ

��,= =

Δ τ( ) α12sn 2Kτ k,( )2 α2+ .=

εw gi( ) VG, M YAG td

a

0.5

∫ S, M ε ΦBC td

0

0.5

∫ H, ΦAF td

0

a

∫ k ΦDF td

0

0.5

∫ ΦBC td

0

0.5

∫–+ 0,= = = =

–4

–5

4000–6

300200100

y

x

Fig. 6. A segment of the flow pattern near the flex point E of the seepage line with base values ε = 0.4, VG = 0.35, T = 7,and S = 5.8, and H =3 and H0 = 1 (the lower curve) and H = 1 and H0 = 3 (the upper curve).

–5.5

–9.5

0.5 0.80.2–1.5

Q

ε

–9

–18

0.20 0.350.050

Q

ε

VG

(a) (b)

Fig. 7. The quantity Q versus ε and VG.

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BERESLAVSKII et al.

(the last dimensionless relation follows directly from considering the velocity potential on the intervals AF,DF, and BC). Then, we calculate the coordinates of the points of the free surface

(15)

Making t = 0.5 in the last equations, we obtain the desired seepage characteristics:

(16)

It is worth noting that the numerical solution of the problem is realized in a bounded variation rangeof the physical parameters of the model.

In Fig. 6 we show the seepage lines whose coordinates are calculated by formulas (15) with base valuesε = 0.4, VG = 0.35, T = 7, S = 5.8 and with H = 3 and H0 = 1 (the lower curve) and H = 1 and H0 = 3 (theupper curve). The calculation of the effect of generating physical parameters ε, VG, T, S, H, and H0 ondimensions d and L are illustrated in Tables 4 and 5, which consist of two sections corresponding to twobase cases: when H > H0 (the upper part of the tables) and when H < H0 (the lower part). Figure 7 illustratesquantity Q versus these parameters.

The following conclusions can be made from Tables 4 and 5 and Fig. 7.

Increasing the inflow seepage intensity, incident velocity, and both pressure heads and decreasing thedepth of the formation and the length of the cutoff wall results in a reduction of quantity d or, which is thesame, in an increase in the ordinate of point F at which the seepage line comes out from the cutoff wall.The parameter which has the greatest effect on quantity d is the depth of formation T: it is seen from Table 4that if parameter T increases by factor 1.1 only, then, the value of d increases by 23.5%. From the datagiven in the right�hand sections of Tables 4 and 5, it is seen that quantity d varies linearly in T and H0,which is natural from a physical point of view.

Of special interest is, first, the position of point C lying on the boundary of some impermeable abut�ment and on the left margin of the underlying aquifer, and second, the associated behavior of quantity L.If parameters VG and H increase and if ε and S decreases, then width L increases. This being so, varyingthe inflow seepage intensity and incident velocity changes width L by a factor of 26.7 and 16.6, respec�tively. On the other hand, varying parameters T and H0 results in the same values L = 98.60 when H > H0

xDF t( ) Mk XDF td( ), yDF t( )

0

t

∫ d– Mk YDF td( ), 0 t 0.5.≤ ≤

0

t

∫+= =

d T H0– M ε ΦDF td , Q

0

0.5

∫– M ε ΨAB t, Ld

0

a

∫ M XDF td

0

0.5 δ–

∫ XCD td

0

0.5ρ δ–

∫–⎝ ⎠⎜ ⎟⎛ ⎞

δ 0→lim .= = =

Table 4.

H, H0 ε d L VG d L T d

H = 3H0 = 1

0.2 5.130 88.10 0.05 5.324 5.932 6.9 4.691

0.5 4.993 7.686 0.20 5.014 16.84 7.5 5.291

0.8 3.915 3.299 0.35 4.791 98.59 8.0 5.791

H = 1H0 = 3

0.2 3.707 27.94 0.05 3.774 1.893 6.90 3.497

0.5 3.695 2.216 0.20 3.706 4.714 7.05 3.647

0.8 3.374 0.951 0.35 3.597 30.83 7.20 3.797

Table 5.

H, H0 S d L H d L H0 d

H = 3H0 = 1

4.8 4.788 99.0 2.0 5.204 64.70 0.5 5.291

5.4 4.790 98.7 3.5 4.586 115.4 0.8 4.991

5.9 4.792 98.6 5.0 3.973 165.5 1.1 4.691

H = 1H0 = 3

3.60 3.595 31.0 1.0 3.597 30.8 2.80 3.797

3.75 3.596 30.9 2.0 3.169 64.0 2.95 3.647

3.90 3.598 30.7 2.5 2.958 80.3 3.10 3.497



and L = 30.83 when H0 > H, so that the effect of the depth of the formation and the pressure head on theunderlying aquifer on the coordinate of point C is not very pronounced.

The calculations show, as in the limit case H0 = 0, that varying all the physical parameters of the modelsalso results in insignificant fluctuations of the seepage flow rate (up to a factor of 1–1.3), while, at the sametime, the linear dependence of quantity Q on the parameters being varied is observed.

Comparing the results of the calculation of quantity d for the same values of the parameters ε, VG, andT shows that in the case when H > H0, the uplift on the water face downstream of the cutoff wall exceedsthe values of d for the case H0 > H by 30–40% (and even as much as 64% when H varies). This is even morepronounced when comparing width L: in the case H > H0, a change in the parameters ε and VG makes Lincrease by 213–215% against the corresponding values in the case H0 > H.

An important role in forming the current is played by the inflow seepage and pressure heads in the walland in the underlying aquifer. Above, we have shown that increasing the inflow seepage and the pressurehead in the underlying aquifer results in a decrease of d and, hence, is accompanied by an uplift of the freesurface. Also, as the calculations show, the flex point E, when moving along the border to the left,approaches the cutoff wall; on the right side, the seepage line becomes steadily flatter and actuallybecomes a horizontal boundary (Fig. 6); this is the signature of incipient drowning.

This therefore brings out an upward effect of the inflow seepage and pressure flow made by the watersof the underlying aquifer with respect to the seepage under the cutoff wall. This is quite a common case inthe practice of hydraulic engineering and irrigated farming when the pressure head at the abutment of cov�ering blankets increases due to the regular creeping of the infiltration moisture, given its insufficient nat�ural withdrawal. The drowning becomes even more developed, followed by a subsequent increase of theparameters ε and H0, but, now within the framework of a different seepage pattern. Here, one may expecta second flex point on the free surface followed by the formation of a groundwater hill around the cutoffwall, just as this happens in similar seepage patterns from channels (see [11,12]).

The limiting flow case is H0 = 0 (no backwater condition). Analysis shows that if all the physical param�eters of the pattern are fixed, then, as the pressure head in the lower highly permeable aquifer decreases,the flex point on the free surface E moves along the border in the direction of point F to eventually mergewith it in the limit when λ = λ* = 0.5ρ. With these values of λ, the right�hand side’s half�disk |w – 0.5(1 +ε)i| < 0.5(1 – ε) drops out in the domain of complex velocity w, and in the plane of flow z, the seepage linebecomes steadily flatter at point F to reach the roof of the underlying aquifer at a right angle at the samepoint. Since, for λ = λ*, we have (see [9])

and using the available relations between the elliptic and theta functions

we can rewrite expressions (12) and (13) as follows

(17)

(18)

Since, in the limiting case λ = λ* = arth(VG/ε1/2)/π, we have

(19)

Formulas (17)–(19) agree with the corresponding formulas (10)–(12) of [1].

REFERENCES

1. E. N. Bereslavskii, L. A. Aleksandrova, and E. V. Pesterev, “Modeling of Some Seepage Flows under HydraulicStructures,” Matem. Model. 22 (6), 27–37 (2010).

2. I. N. Kochina and P. Ya. Polubarinova�Kochina, “On Application of Smooth Contours of Abutments ofHydraulic Structures,” Prikl. Mat. Mekh. 16, 57–66 (1952).

3. P. Ya. Polubarinova�Kochina, Theory of Motion of Underground Water (Gostekhizdat, Moscow, 1952) [in Rus�sian].

ϑ2 τ 0.5iρ+( ) 1

q4��e πτ i– ϑ3 τ( ), ϑ2 τ 0.5iρ–( ) 1

q4��eπτiϑ3 τ( ),= =

ϑ1 τ( ) ksn 2Kτ k,( )ϑ0 τ( ), ϑ3 τ( ) 1

k '��dn 2Kτ k,( )ϑ0 τ( ),= =

w τ( ) ε πτ,tan=

dωdτ�� εM πτsin

sn 2Kτ k,( )Δ τ( )��, dz

dτ�� M πτcos

sn 2Kτ k,( )Δτ�� .= =

ρ K 'K�� 2arth ε

π��, g

arth VG/ ε( )π

�� .= = =

628


BERESLAVSKII et al.

4. W. von Koppenfels and F. Stallmann, Praxis der Konformen Abbildung (Springer, Berlin, 1959; Inostrannaya Lit�eratura, Moscow, 1963).

5. E. N. Bereslavskii, “Conformal Mapping of Some Circular Polygons on a Rectangle,” Izv. Vyssh. Uchebn.Zaved., Mat., No. 5, 3–7 (1980).

6. E. N. Bereslavskii, “On Differential Equations of the Fuchs Class Associated with the Conformal Mapping ofCircular Polygons in Polar Grids,” Differ. Uravn. 33 (3), 296–301 (1997).

7. E. N. Bereslavskii, “Closed�Form Integration of Some Fuchsian�Class Differential Equations Found in Hydro�and Aeromechanics,” Dokl. Math. 80 (2), 706–709 (2009).

8. E. N. Bereslavskii, “On Closed�Form Integration of Some Fuchsian Differential Equations Related to a Con�formal Mapping of Circular Pentagons with a Cut,” Differential Equations 46 (4), 459–466 (2010).

9. I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series and Products (Nauka, Moscow, 1971; AcademicPress, Boston, 1994).

10. V. V. Golubev, Lectures on Analytical Theory of Differential Equations (Gostekhizdat, Moscow, 1950) [in Rus�sian].

11. E. N. Bereslavskii, “On the Effect of Impermeable Inclusion in Underlying Highly Permeable Formation onGround Water Regime in Watered Soil Layer,” Prikl. Mekh. Tekh. Fiz., No. 3, 3–5 (1986).

12. E. N. Bereslavskii, “On the Effect of Impermeable Inclusion in Watered Soil Layer to a Channel Seepage,” Izv.Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5, 71–76 (1990).

Documents

On ground water seepage under hydraulic structures