ELSEVIER

Advances in Water Resources, Vol. 20, No. 4, pp. 207-216, 1997 Copyright 0 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved PII: SO309-1708(96)00020-6 0309-1708/97/$17.00+0.00

Groundwater flow with free boundaries using the hodograph method

Mark Bakker Department of Civil Engineering, University of Minnesota, 500 Pillsbury Drive S.E., Minneapolis, MN 55455, USA

(Received 7 August 1995; revised received 29 April 1996)

A new procedure is presented to obtain exact solutions to groundwater flow problems with free boundaries in the vertical plane. The solution procedure makes use of the hodograph method in combination with conforrnal mapping. The complex discharge function and the reference function are used as auxiliary functions. The function that maps the upper half plane onto the domain in the complex discharge plane is obtained by integration of the differential equation of Schwarz. The final solution is a linear combination of infinite series and consists of two functions: the conformal map of the upper half plane onto the physical plane and onto the complex potential plane. As an example, the problem of flow over a horizontal base to a straight seepage face is solved. Flow nets are presented for two inclinations of the seepage face and rules are derived for the specification of boundary conditions along seepage faces in Dupuit-Forchheimer models. Copyright 0 1996 Elsevier Science Ltd

Key words: groundwater, hodograph method, conformal mapping, free boundary problems, seepage face.

INTRODUCTION

A new procedure is presented for solving two- dimensional groundwater flow problems with free boundaries; the procedure is based on the hodograph method in combination with conformal mapping. As an illustration of the procedure, the problem of unconfined flow to an embankment without tailwater is solved. The resulting exact solution is used to derive rules for the placement of boundary conditions along seepage faces in Dupuit-Forchheimer models.

Two-dimensional groundwater flow problems in the vertical plane are referred to as free boundary problems if the boundary includes a phreatic surface or the interface between two fluids of different densities because these boundaries are free to adjust their position to the boundary conditions. Problems with free bound- aries may be solved in an approximate manner by application of the finite element and finite difference methods (Crank7), series solutions to Laplace’s equa- tion (Read and Volker14), or the hodograph method in combination with boundary integral techniques (Detour- nay and Strack’). Exact solutions may be obtained by the use of the hodograph method in combination with conformal mapping (Aravin and Numerov’ , Bear3, Harr”, Polubarinova-Kochina13, Strack”).

The problem is formulated in terms of a complex

207

variable z and a complex potential s1. A solution is sought in the form z(c) and G(C), where 5 is a parameter defined in the upper half plane (Xc > 0). Two auxiliary functions are selected that have known boundaries and that can be used to obtain z(c) and fit(<). The complex discharge function (as introduced by Hamel” and Davison8) and the reference function (as introduced by Strack and Asgian16) are used as the auxiliary functions in this paper. These two auxiliary functions may be used to solve, in principle, problems involving seven different boundary types: a phreatic surface with or without infiltration, the interface between two fluids of different densities, and straight equipotentials, streamlines, see- page faces, and outflow faces (Strack”).

The domain in the complex discharge plane is bounded by straight segments and circular arcs for the aforementioned seven boundary types. The function that maps the upper half plane onto such a domain may be obtained by integration of the differential equation of Schwarz (Van Koppenfels and Stallmann17 and Nehari12). This differential equation contains unknown parameters for domains with more than three corner points-the determination of these parameters is known as the parameter problem (Van Koppenfels and Stall- mann17). The procedure outlined in this paper may be used to solve the parameter problem.

The reference function is a combination of the

208 M. Bakker

derivatives of z and the complex discharge function with respect to the reference parameters C. The domain in the plane of the reference function is bounded by straight segments that lie on lines through the origin for the aforementioned boundary types. The upper half plane may be mapped onto such a domain by application of the mapping with piecewise constant argument (see Strack”).

Problems with three corner points may be solved alternatively by rewriting the differential equation of Schwarz into the hypergeometric differential equation and by using hypergeometric series to obtain a solution (Strack and Asgian16); this approach cannot be applied to problems with more corner points. Solutions to problems with more corner points may be obtained by application of the method described by Polubarinova- Kochina,13 which is based on the analytic theory of linear differential equations; her method has been applied by, for example, Bereslavskii415

This paper is divided into two parts. In the first part, the new solution procedure is outlined. This part includes a review of the hodograph method and the differential equation of Schwarz, a derivation of expressions for z(c) and n(c) and a discussion of how to determine the constants in the solution. In the second part, the procedure is applied to the problem of flow to a straight seepage face. The presented solution is used to derive rules for the specification of boundary conditions along seepage faces in Dupuit-Forchheimer models.

THE HODOGRAPH METHOD

Groundwater flow problems may be formulated in terms of a complex variable z = x + iy (x and y are Cartesian coordinates) and a complex potential

R=@++i9 (1) where @ is the discharge potential and 9 the stream function. The discharge potential for flow in a vertical plane of width B normal to the direction of flow is (Strack”)

Q=kBcj (4 where k is the hydraulic conductivity and 4 the piezometric head. The discharge potential and stream function are related through the Cauchy-Riemann conditions

a@ a* aa aq -=_ _-- ax ay &- ai (3)

The complex potential R is an analytic function of z for aquifers with a constant hydraulic conductivity.

The components QX and QY of the discharge vector, the amount of flow over a width B of aquifer, are

and the complex discharge W is

W+Q,-jQ,

The complex conjugate of the complex discharge is the hodograph

w= Q,+iQ,. (6)

The hodograph method may be applied in combina- tion with conformal mapping to obtain exact solutions to free boundary problems. The hodograph method makes use of two auxiliary functions, which in this paper are the complex discharge function W and the reference function R, introduced by Strack and Asgiani6

R(C) = W’(C)14C)12 (7) where the prime (‘) indicates differentiation with respect to the reference parameter < = < + in, defined in the upper half plane (YC 2 0). The upper half plane may be mapped conformally onto the domains in the W- and R-planes to obtain expressions for W = W(c) and R = R(C). Once these functions are determined, an expression for z’(C) is obtained from equation (7) as

z’(C) = R”2(<)( W’)-1’2(<) (8) and an expression for n’(c) from equation (5) and by using the chain rule

df2 dR dz (2’(C) = z = x x = - W(<)z’(<).

Integration of equations (8) and (9) gives the desired functions z(c) and a(<).

The difficulty in the application of the hodograph method lies in the determination of the mapping of the upper half plane onto the domain in the W-plane which is bounded by straight segments and circular arcs.l’ For a limited number of problems the boundary of the domain in the W-plane may be mapped onto a domain bounded by straight segments only; the upper half plane may be mapped onto such a domain with the Schwarz-Christoffel integral.‘3Y’5 If this is not possible, the differential equation of Schwarz has to be used.

THE DIFFERENTIAL EQUATION OF SCHWARZ

The conformal mapping of the upper half plane onto a domain bounded by straight segments and circular arcs may be obtained by integration of the differential equation of Schwarz (Von Koppenfels and Stallmann,” p. 116)

$ln W’ - 1 -&In W’ 2= S(c) [ I

(10)

where S(c) is the differential expression of Schwarz. S(c)

Groundwater flow with free boundaries 209

is defined for a domain with N corner points as

N-l

N-l

‘S(c) = 5 (< :< [ 1 K,v - CK” CN-3

2 ”

)2 +- N_;=’

j$z-E,)

N-4

gp’ /

N-l (11)

v=l

The parameters & (V = 1,2,. . . , N - 1) correspond to corner points 1 through N - 1, respectively. Two parameters <, may be selected arbitrarily; corner point N corresponds to < = & = 00. The parameters Pj (j=O,l,..., N - 4) are referred to as the accessory parameters and the constants K, are defined as

(12)

where 7rk, is the change in argument at corner point v of a complex increment dW while moving along the boundary in a positive sense, that is with the domain to the left.

The function p is introduced as

P = (pf/7/2 (13)

Combination of equation (13) with the differential equation of Schwarz (10) gives, after differentiation and combining terms,

/_Y+;s~=o (14)

This is a second order linear ordinary differential equation with regular singular points at < = <, (V = 1,2,. . , N). Solutions to such a differential equation are obtained by the use of power series (Boyce and Diprima6, Chap. 4).

The general solution p = ( W’)-“2 to equation (14) is a linear combination of two linearly independent solutions to the differential equation, labeled as pa and

Ph

( W’fm-” = p = A*pa + B*,LL/, (15)

where A’ and B* are complex constants. The solutions pL, and & are power series and are given in the appendix. The complex discharge function W may be obtained from equation (15) as

WE dc+w .I’ (d2 o (16)

where W, is a complex constant. For a differential equation of the form (14), the following relationship

holds (Boyce and Diprima6, section 3.2).

(17)

provided that &, is linearly independent of p (D* is a constant). Combination of equation (15) (16) and (17) gives

w= D*& A*PL, + B”Pb

+ w,. (18)

THE FUNCTIONS z(C) AND Q(C)

Expression (15) for ( W’)-1’2 may be used in combina- tion with the reference function to obtain an expression for z’ (using equation (8)). The square root of the reference function, given by Strack and Asgian,16 yields

N-l

R’12 = E l--(< _ [n)-in-kn12 = ,L$ (19) n=l

where E is a complex constant, r/t, is the change in argument of a complex increment dz at corner point n while moving along the boundary in the z-plane in a positive sense, and p = Rli2/E.

Substitution of equation (15) for ( W’)-1’2 and equation (19) for R l/2 in equation (8) for z’ gives

z’ = R”2( W’)--“2 = Ep[A*pL, + B*&,]

= AX, + BXL,

where

(20)

&I = PPa Xb = k+b (21)

and A = EA*, B = EB*. The functions X, and Xb may again be written as power series, as is shown in the appendix.

An expression for R’ is obtained by substitution of equation (18) for W and equation (20) for z’ in equation

(9)

fll= -wz’ = - D+iJb ASPS, + B*P~

+ Wo I (A*pLn -c B*h)&

= - W,,AX, - (D + WoB)‘\b (22)

where D = ED*. Two new constants are introduced

i=-WOA i = -(D + WoB) (23)

so that

0’ = kx, + &+,. (24)

Note that both z’, equation (20) and a’, equation (24), are linear combinations of the power series X, and &.

Integration of equations (20) and (24) gives

z=AA,+BAb+C (25)

210 M. Bakker

where C and c are constants of integration, and A, and hb are obtained from integration of the series X, and X,

A, = s

X,d< hb = s

Xbd<. (26)

This integration may be carried out term by term, as is shown in the appendix.

Expressions (25) for z(c) and n(C) are linear combinations of the power series A, and Ab. The series A, and Ab are expanded about a point, say < = in, and have a circle of convergence centered at < = <n with a radius equal to the distance between (’ = 5, and the nearest singular point. A series expansion converges only inside its circle of convergence, so that a complete solution consists of several series expansions in such a way that the union of the circles of convergence of all expansions covers the whole upper half plane. A power series about a point < = &, on the real axis will be labeled with the superscript(“) and the constants in the expression with the index “. The expansion of z about < = & will thus be written as

z@) = A A(“) + BvAf’ + C ” (I “’ (27)

A solution that consists of N expansions of z(C) and Q(c) contains 6N unknown complex constants: AV,BV,CV,&,hy and c, (v= 1,2 ,..., N). Only a few of these constants may be obtained by applying the boundary conditions directly. A procedure has been developed to determine all constants that may not be obtained from the boundary conditions by the use of analytic continuation.

ANALYTIC CONTINUATION

It will be shown in what follows that the unknown constants of an expansion about one point, say point n, may be expressed in terms of the unknown constants of the expansion about an adjacent point, say point n + 1. The derivation is presented here for expansions of z’, but is equally valid for expansions of 0’.

Consider expansions of z’ about < = &, and C = &+I that have circles of convergence that overlap (see Fig. 1). The two expansions may be set equal at the points C = 5, and C = cP where both converge

(z’)‘““‘(<,) = (z’)‘“‘(Cp)

(z’)‘““‘(~J = (z’)‘“‘(&). (28)

Substitution of equation (20) for z’ gives the following set of equations (where X, = &J&J, A, = UC,), etc.)

The expansion about < = <,, is said to be continued analytically into the domain where the expansion about

n n+l Fig. 1. Circles of convergence of two adjacent series expansions.

c = Fn+l converges when the constants A,+, are chosen such that they fulfill equation (29). B n+l may be expressed in terms of A,, and B,

A n+l - - *I IA, + ~124

B ,,+I = *21An + ~22%

and &+I A n+~ and

(30)

The constants wll, w12, w21 and w22 may be obtained by application of Cramer’s rule (provided that cp and & are two separate points.)

In a similar way, the constants of the expansion of z’ about C = &+2 (An+2 and Bn+2) may be expressed in terms of A,+, and B,,, . But since A,+1 and B,,,, are already expressed in terms of A,, and B, (equation (30)), so may An+2 and Bn+2. This procedure may be repeated until all coefficients A, and B, (v = 1,2,. , N; u # n) are expressed in terms of A, and B,,. The important result is that A, and B, have to be determined for only one value of v from the boundary conditions; all other values may be obtained from analytic continuation.

The values of C, (V = 1, . . . , N) may be obtained in a similar way. If, for example, C,, is known, the value of C n+l may be obtained by setting the expansions of z about < = [,, and < = cn+, equal at C = Cp where both expansions converge. The resulting expression contains only one unknown (C,,,) which may be determined.

It is noted that the differential expression of Schwarz (11) contains unknown parameters for problems with more than three corner points; the mapping function depends on these parameters in a non-linear way. The determination of these parameters is commonly known as the parameter problem. The parameter problem may be solved iteratively by application of the procedure presented in the foregoing. Starting with a set of initial values, the parameters are adjusted until all boundary con- ditions are met. This approach has been applied success- fully to a problem with five corner points by Bakker.’

PLANE CORNER POINTS

Before presenting an example, the mapping function near a plane corner point (the intersection of two

Groundwater flow with free boundaries

straight segments) is investigated. The analysis is presented here for some generic function F; the upper half plane is mapped onto the domain in the F-plane. Point v is a plane corner point in the F-plane and the change in argument of F’ at comer point v is 7~ (see Fig. 2). Von Koppenfels and Stallmann” (equation 12.1.11) show that F’ may be represented near C = E,, corresponding to point v, by

F’ = (C - L-WI) (31) where G is a function that is analytic at C = &. Note that the argument of (C - 5,) decreases by 7r at C = c, while moving in the positive direction along the e-axis: the argument of F’ will thus increase by TIC, as asserted. G may be expanded in a Taylor series about C = 5” with coefficients “/n

F’ = (C - <uy>-“[ro + YI (< - &) + 72(C - tJ2 + . . .I.

(32)

Similarly, if point v corresponds to infinity, F’ may be represented near infinity by

F’ = <-‘2-“‘[“/o + y~(-’ + -y2C2 + . ] . . . (33)

Hence, F’ may be represented near plane corner point v by a Taylor series multiplied by a term (C - &,-” (or a Laurent series multiplied by a term c-(2-n) if Y = N) where 7~ is the change in argument of F’ at point u.

This result may be applied to z’ for example, as follows. Consider plane comer point Y in the z-plane, corresponding to C = &,,; the change in argument of z’ at v is k,. According to equation (32), z’ may be represented near < = 5, by a term (C - &)-kU multiplied by a Taylor series in terms of (C - &,). Inspection of X, and Xb, equation (A4), shows that X, has this form, so that z’ is a multiple of X, only, rather than a linear combination of X, and Xb, equation (20), and thus

B, = 0. (34)

FLOW TO A STRAIGHT SEEPAGE FACE

Consider the problem of groundwater flow over a horizontal impermeable base towards an embankment without tailwater: a straight seepage face (see Fig. 3(a)). The seepage face makes an angle a with the base, the aquifer is semi-infinite, and there is no infiltration. The

Fig. 2. A plane corner point.

Fig. 3. Flow to a seepage face (a) z-plane; (b) R-plane.

origin of a Cartesian x, y coordinate system is chosen at the intersection of the seepage face and the base, and is numbered point 1. Point 2 is at the intersection of the water table and the seepage face; point 3 is at infinity. The boundary conditions are as follows. The base of the aquifer is impermeable. The elevations of the seepage face and the phreatic surface are equal to the hydraulic head $J along these boundaries. In addition, the phreatic surface is a streamline. The total amount of flow over a width B of aquifer normal to the plane of flow towards the seepage face is Qo. The length of the seepage face is unknown a priori, as is the location of the phreatic surface. The phreatic surface is a _free boundary in the z-plane. The change in argument rk, of z’ at each corner point is obtained from Fig 3(a)

k, = 1 - a/7r It2 = 0 I& = 1. (35)

The shape of the domain in the o-plane is shown in Fig. 3(b). The stream function is chosen zero along the base (segment 3-l), so that q equals -Q. along the phreatic surface (segment 2-3). The position of the seepage face (segment l-2) is unknown in the R-plane.

The boundary of the domain in the hodograph plane (w) is constructed by application of the rules given by, for example, Strack,15 Chapt. 7. The real and imaginary axes correspond to Q, and Q,,, respectively. The boundary consists of three segments. The horizontal impermeable base, segment 3- 1, corresponds to part of the real axis (Q, = 0). The seepage face, segment 1-2, corresponds to a straight line normal to the seepage face and through Q, = -kB. The phreatic surface, segment 2-3, corresponds to part of the circle centered at Q,, = -kB/2 and through the origin. Connection of the three segments of the boundary results in the domain shown in Fig. 4(a). The domain in the W-plane is

212 M. Bakker

obtained by reflection through the real axis and is shown in Fig. 4(b). The boundary in the W-plane consists of straight segments and circular arcs, as asserted. The change in argument nk, of W’ at each corner point is obtained from Fig. 4(b)

kl = a/~ + l/2 kZ=a/7r+1/2 k, = 1.

(36)

The c-plane is chosen such that point 1 corresponds to <=<i=-1, point 2 to <=&=+l and point 3 to C = 6s = co. Series expansions for z(c) and Q(C) are derived about < = &, C = &, and C = Es. The expansions about < = & and C = & have a radius of convergence equal to 2 (the distance between Et and .$); the asymptotic expansion converges outside the unit circle. The union of the circles of convergence of these series covers the whole upper half plane as is shown in Fig. 5.

Expansions of z and R are obtained from equation (25) and may be written as

1

z(l) = At&!) + BIAf) + Cl for I<+ 11 < 2

z = .z(~) = AZ&?) + B2AF’ + C2 for I< - 11 < 2

zc3) = A3Af) + B3Ar3) + C3 for 151 > 1

(37)

R(i) = AlAi) +&A!’ + C;, for I< + 11 < 2

St(2)=~2A~)+B2A~)+C2 for)<-1(<2.

Q(3) = A3A(,3) + B3Ari + C3 for I<[ > 1

(38)

kB

Fig. 4. (a) The hodograph plane; (b) the W-plane.

(39)

Fig. 5. Circles of convergence of the three series expansions.

The series AL’), A!), AL*’ and Af) are obtained by following the procedure outlined in the Appendix (see equations (A9))

Ai’) = (C + l)a/n Ta!‘)(< + 1)” n=O

Ab” = (< + 1)‘12 Tb!i’(c + l)n n=O

Ai2) = (<- l)~a~*)(<- 1)” n=O

AC21 = ([ _ 1)3/*-a/” h T@(C - 1)“.

n=O

(40)

Point 3 is a special point, because k3 = 1. The series Xp) and Xf’ are obtained from equations (A8), using equations (35) and (36)

XC) = <-* 2 a$,<-” + Xf’ In < n=O

AK) = c-i 5pc-~.

(41)

II=0

Ai3’ and Ar’ are obtained by integration of these expansions term by term. Integration by parts and combining terms gives

Ai31 = J

@)dC = I-’ 2 n=O _$+ +@Y(O

cc p(3) _ In(C)C*C-” _ ~!$!.(-a

II=1 ?%I

Ap’ = s

fx $3, Xf)dC = ,$‘ln(<) - xe[-fl.

il=l

(42)

The constants A,, B,, C,, j,, &,, c, (v = 1,2,3) in

Groundwater flow with free boundaries 213

equations (37) and (38) have to be determined for one value of v only. All remaining constants may then be obtained from analytic continuation, as previously explained.

DETERMINATION OF THE CONSTANTS

The constants in the expansions of R are determined first. Point 3 is a plane comer point in the a-plane and corresponds to C = 00; the change in argument of R’ at point 3 is fl (see Fig. 3(b)). Hence, R’ may be represented near < = 0~) by a term c-i multiplied by a Laurent series (compare equation (33)). Inspection of equation (41) shows that Xf’ has this form, so that

As = 0. (43)

The value of 8, may be determined from the behavior of fl at infinity. The stream function jumps by Q0 at infinity, as may be seen from Fig. 3(b): 9 equals -Qe along the phreatic surface (2-3), and 0 along the base (3-l). The imaginary part of the logarithm in RF), equation (42) jumps as well: along segment (2-3) 9(ln C) is 0 and along segment (3-l) Y(ln [) is 7r. Hence, Q(3) jumps by iQO at infinity if Bs is chosen as

B3 = Q&d3’) 0 . (44)

The value of c, may be determined by substitution of C = t1 = -1 in R(l), which corresponds to R = 0 (see Fig. 3(b))

n(“(< = -1) ZZ c, = 0. (45)

The values of k3, & and ci are now determined; the remaining constants in the expansions of R may be determined from analytic continuation, as previously explained.

The constants in the expansions of z will be determined next. Point 1 is a plane corner point in the z-plane, so that (compare equation (34))

B, = 0. (46) The value of Ai is obtained from equation (23)

Al = -Wo.1A, (47)

and Wo,, may be obtained from the expansion of W about C = <i (see equation (18))

WC’) = D;pj)) &p + B*Jl) + wo.l.

I h

(48)

Substitution of equations (60) for pL, and & and some basic series manipulation gives

W(‘)(< = -1) = W,,, = kB/tan(r/2 - a) (49)

where is it noted that C = El = -1 corresponds to W = kB/tan(n/2 - a), as may be seen from Fig. 4(b).

Combination of equations (47) and (49) gives

Ai = -Ai tan(n/2 - cx)/kB (50)

The value of c, may be determined by substitution of < = Ii = -1 in z(l), equation (37)

z(‘)(C = -1) = Ci = 0 (51)

since the origin in the z-plane is chosen at point 1. The values of Ai, Bi, and Ci are given by equations

(50), (46), and (51), respectively; the remaining constants in the expansions of z may be determined from analytic continuation, as previously explained. The constants in the expansions of z and R are now determined and the problem is solved. The solution has been implemented in a FORTRAN program; the series were truncated after 50 terms. Flow nets near the seepage face are presented for a = 30” and cr = 60” in Fig. 6.

SEEPAGE FACES IN DUPUIT-FORCHHEIMER MODELS

The presented exact solution is used to derive a rule for the placement of boundary conditions along seepage faces in Dupuit-Forchheimer models.

For unconfined flow, the total discharge Qx flowing in the x-direction per unit width of aquifer is

where h is the elevation of the groundwater table, qx the specific discharge in x-direction, and where use is made of Darcy’s law. Application of Leibniz’s rule gives (for a constant hydraulic conductivity)

(53)

The total discharge in the x-direction may now be written as minus the gradient of a potential U (using that 4(x, h) = h)

Qx+

where

U= hk4dy-fkh2. s 0

Finally, application of continuity of flow gives

dQx d2U -zz----_ -N(x) dx dx2 (56)

where N(x) is a source term. Solutions to groundwater flow problems may be obtained by solving equation (56) with the appropriate boundary conditions. Exact solutions for Qx are obtained if boundary conditions in terms of head are transformed to potentials using equation (55).

214 M. Bakker

a = 300

Fig. 6. Flow nets for (Y = 30” and LY = 60”.

For regional flow it is often assumed that the head is constant over the vertical and equal to the elevation h of the water table (the Dupuit-Forchheimer assumption); substitution of 4 = h in equation (55) gives

U = ;kh2. (57)

The question arises as to what condition should be specified along a seepage face. The potential along a vertical seepage face may be obtained by substitution of 4 = y in equation (55)

=O.

Hence, h = 0 should be specified along a straight seepage face, so that with equation (57) U = 0.

A rule for the specification of heads along seepage faces that are not vertical is derived from the exact solu- tion presented in this paper. The integration of equation (55) is carried out analytically and implemented in a

i t*k X

!a R

Fig. 7. The potential near a seepage face.

FORTRAN program. Left of the seepage face the total discharge Qx is equal to Qa and the function U(x) is a straight line (see Fig. 7):

U(x) = -Q,,(x + I) for x < -Lx. (59)

Hence, h = 0 (corresponding to U = 0) should be specified at x = -1; 1 may be obtained from Fig. 8 where 1/(L cos a:) and kBL cos a/Q,, are plotted vs a. Implementation of this rule requires the user to estimate L or Q0 to obtain an initial value of 1; the value of I may then be updated during the modeling process.

CONCLUSION

A general procedure has been presented to obtain exact solutions to groundwater flow problems with free boundaries using the hodograph method. A solution was obtained in the form z(C) and a(<). It was shown that z(C) and a(<) are, at any point in the aquifer, a linear combination of two infinite power series A,(c) and hb(C) plus a constant; the series have to be truncated for implementation in a computer program. A complete solution consists of a number of power series, each expanded about a different point, such that the union of the circles of convergence of all series covers the entire upper half plane. The constants in the solutions have to be determined for the expansion about one point only, since the constants of an expansion about a neighboring point may be obtained by analytic continuation.

kBLcosa

QO

0.46

0.46

0.44

0.42 I

Lcosa 0.40

038

0.36

Fig. 8. Graph of kBL cos a/Q0 (solid) and I/(L cos a) (dashed) vs ff.

Groundwater JEow with free boundaries 215

It was indicated that the presented procedure may also be used if the problem has more than three corner points and the differential equation of Schwarz contains unknown parameters. The solution depends on these parameters in a non-linear way and the parameter problem must be solved in an iterative fashion. Further research is needed to develop an algorithm to solve the parameter problem efficiently for problems with many corner points.

APPENDIX

ACKNOWLEDGEMENT

The author thanks Otto Strack for his suggestions and support.

In this appendix, a procedure is outlined to obtain the coefficients of the series A, and I&, (26). A, and Alb are the integrals of the functions X, = ,o~~ and Xb = P&, (21), where pL, and & are two linearly independent solutions to the differential equation of Schwarz (14). pa and pb are obtained as series expansions about a singular point < = & as (Von Koppenfels and Stallmanni7, Chap. 12)

lla = (C - &JkU’2[ ao+al(~-~~;)+a2(t-~,)2+...l

Pa = (5 - wku’2h + TIC< - L/l + T2(< - 5J2 + ‘. .I

REFERENCES

+ -wa WC - E,) (Al) where the constant y may only be unequal to zero if k, is an integer; for the case that k, = 1, y and T,, equal 1 and 0, respectively. Recurrence relations for the coefficients c,, and r,, (n = 0, 1, . . .) may be obtained by substitution of pa and &, respectively, in the differential equation (14) and setting the sum of coefficients of like powers of (< - <,,) equal to zero. Two linear independent solutions near < = & = co are

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

Aravin, V. I. & Numerov, S. N. Theory of Fluid Flow in Undeformable Porous Media. Danien Davey, New York, 1965. Bakker, M., Two-dimensional groundwater flow with free boundaries using the hodograph method, Ph.D. Thesis, Department of Civil Engineering, University of Minne- sota, MN, 1994. Bear, J. Dynamics of flow in porous media. American Elsevier, New York, 1972. Bereslavskii, E. N. & Panasenko, L. A. Fresh water lenses formed by seepage from irrigation channels, Fluid Dynamics, 1990, 25, 580-4. Bereslavskii, E. N. Effect of an impermeable inclusion in an irrigated soil layer on seepage from a channel. Fluid Dynamics, 1990, 25, 708-13. Boyce, W. E. & Diprima, R. C. Elementary Dtyerential Equations and Boundary Value Problems, 3rd edn. Wiley, New York, 1977. Crank, J. Free and Moving Boundary Problems. Oxford University Press, New York, 1984. Davison, B. B. On the steady two-dimensional motion of ground-water flow with a free surface. Phil Mag., 1936,21, 881-903. Detournay, C., & Strack, 0. D. L. A new approximate technique for the hodograph method in groundwater flow and its application to coastal aquifers. Water Resour. Res., 1988, 24, 1471-81. Hamel, G. Ueber Grundwasser. Z. Angew. Math. Mech. 1934, 14, 129-57. Harr, M. E., Groundwater and seepage. McGraw-Hill, New York, 1962. Nehari, Z., Conformal Mapping. Dover Publications, New York, 1975. Polubarinova-Kochina, P. Y. Theory of Groundwater Movement, (translated by J. M. R. De Wiest.) Princeton University Press, Princeton, NJ, 1962. Read, W. W. and R. E. Volker. Series solutions for steady seepage through hillsides with arbitrary flow boundaries. Water Resour. Res., 1993, 29, 2871-80. Strack, 0. D. L. Groundwater Mechanics. Prentice-Hall, Englewood Cliffs, NJ, 1989. Strack, 0. D. L. and Asgian, M. A new function for use in the hodograph method. Water Resour. Res. 1978, 14, 1045558. Von Koppenfels, W. and F. Stalhnann. Praxis der Konformen Abbilbung. Springer, Berlin, 1959.

where y may only be unequal to zero if kN is an integer; for the case that kN = 1, y and cro equal 1 and 0, respectively. Series solutions may also be derived about ordinary points (see Von Koppenfels and Stallmanni7). Recurrence relations for the coefficients o,, and 7, (n = 0, 1, . .) in the series (Al) and (A2) may be found in Bakker’.

The function p, equation (19) may be written as

p = (< _ <“)-Ii& [PO + PI (C - E,) + P2(C - L/l2 + . . .I (A3)

where the coefficients P,, (n = 0, 1, ._.) may be obtained by expanding every term (’ - <n)-kn-kn” (n = 1,2, . . . , N - 1; n # v) in equation (19) in a separate Taylor series, multiplying all N - 2 Taylor series term by term, and gathering terms.

The series X, and Xb about C = $, are obtained by multiplication of p, equation (62). with pa and &,, equations (60)

&I = P/&l

216 M. Bakker

An expression that is valid near < = cc may be obtained by developing p in an asymptotic expansion

P=IOIPotPiC-’ +LK2+...l (A61

where N-l

e = c -rt, - k,/2 ?Z=l

(A7)

Multiplication of equation (A6) with pL, and ph, ‘equations (A2), gives

A, = C ‘+B-kN’2[o!0 + f3ic-i + Q-2 + . .]

+‘dbln< 648)

Ab = c e+kN’2[pl) + p,c-’ + p2<-2 + . .]

where Q, and ,&, (n = 0, 1, . .) are given by equation (A%

Equations (A4) may be integrated term by term for the case that y = 0

A, = J

Ld(C - I,)

= CC - E,)‘-“u.[a0 + ai(C - E,) + a2(5 - &I2 + . . .I

Ah = s MS - <,I = (I - GJ2-iv-ku[b~ + h (S - E,)

+ 62(< - a2 + . . .I (A91

where

a, = a,/(1 - k, + n) b, = ,&,I(2 - k, - k, + n).

(AlO)

Integration of equation (A8) for y = 0 gives

A, = s

j,,dC = C2+e-k,j2 [a0 + al<-’ + a2Ce2 + .]

Ah = .I

XbdC=< ‘+e+kN’2[bo + b,<-’ + b2C-2 -t . . .]

C-411)

where

a, = a,/(2 + 0 - kN/2 - n)

b,=p,/(1+e+kk,/2-n). (Al2)

Integration for the case y # 0 is dealt with on a case by case basis.