Second order linearized theory of free-surface flow in porous media

SECOND ORDER LlNEARIZED THEORY OF FREE-SURFACE FLOW IN POROUS MEDIA

BY G. DAGAN *

A certain analogy exists between the equation of irrotational motion of an ideal liquid bounded by a free- surface and the free-surface flow of a liquid in a porous medium, obeying Darcy's Law.

In both cases the velocity potential is a harmonie function; on the free-surface the kinematic condition is identical. The dynamic one is different: the Bernoulli relation yields equations of a hyperbolic nature (conservative motion) while Darcy's Law yields parabolic equations (dissipative motion).

In both cases the nonlinearity of the potential boundary condition on the free-surface constitutes the main mathematical difficulty of the problem.

However, while in the theory of flow of an ideal liquid (especially water-wave theOl'Y) considerable progress has been achieved by the linearized theory, and approximations of first, second and higher order have been derived, only some first order solutions are lmown in the theOl'y of flow through porous media.

The pm'pose of this paper is to derive the linearized equations of free-surface flow through porous media in a systematic manner, with some applications to the second order theory.

1. The exact equations In this work the saturated flow of an incompressible, homogeneous liquid through an isotropie, homogeneous and non-deformable porous medium will be considered.

The exact equations of free-surface flow are already available in several sources (for example Polubarinova- Kotchina [1952]), and are reproduced here for convenience of reference.

Let p and y be the density and specifie vi'eight of the liquid respectively, [J.-its dynamic viscosity, k-the intrinsic hydraulic conductivity of the porous medium, K = (ky/[J.) its hydraulic conductivity and n-its effec- tive porosity coefficient.

With <p

= -

K [(PlY)

+

^{z] and 'l)(x, y,t),} the velocity potential and the free-surface expression respectively,

the following equations are satisfied:

in the field of flow:

(U)

(1.2)

* Lecturer, Hydraulic Laboratory, Technion-Israel Institute of Techllology.

901

G. DAGAN and:

YJ= -

I f

1 <P(x, y, z, t) on the free-surface z= YJ(x, y, t).

On the other boundaries the more common conditions encountered are:

O<p

=

0

----on

on an impervious boundary:

<P

=

const on the boundary with liquid free of porous medium:

<P

=--

Kz

(1.3)

(lA)

(1.5 )

(I.G) on a seepage face.

Exact solutions to these equations have been achieved only for steady bidimensional flow, mainly by the hodograph method (a brief review of this method is presented by BeaI' and Dagan [1962]).

Il. The derivation of the linearized equations by the approximation of small free-surface disturbances This approximation is analogous to that of infinitesimal-waves. The basic assumption is that both <P and YJ may be expanded in power series of a sma11 parameter E :

<P(x,y, z,t) = <Po(x, y,z)

+

^{E<p]Cr,y,z, t)}

+

EZ<pz (x,y, z,t)

+ ...

YJCx, y,t)

=

YJoCr, y)

+

^{EYJ](x, y,t)}

+

^{EZYJz(x, y, t)}

+ ...

The nature of Eis not specified; obviously, it has the charader of a perturbation.

nonperturbed steady-state.

The substitution of <p from (IU) into (0.1) yields:

\72<p

=

\7z<po

+

^E\72<Pl

+

EZ\7Zq>z

+ ... ⁼

⁰

hence:

OLl)

(11.2)

<Po and YJo represent the

(II.::~)

(lIAI in the flow region.

The condition to be satisfied on the fI-ee-surface wiU result from equations (1.2), (lU) and (11.2).

First, the value of <p and its derivatives on the free-surface may be derived by expansion on the z= 110(x, y) :

1 1

(

o! \ )

( 0 ¹ Z 02

1 0 1

<p(x, y,z, t) Z=7J = <Po Z=7Jo

+

E \ YJ] 0;0 ⁱZ=7J': <p],z=7JO

+ E2,YJZa~lz=7Jo + _~

OZ;.'I,Z='10

+

^{11] 0:]0=710+<P2}

Hence, from (1.3) and (I1.5) by identification:

l :

YJo

= .--_. -

<Po ¹

K iZ="/o

surface

+ .. ,

(11.5)

(II.G)

Expanding <p derivatives in a similar way, substituting in (1.2), it \vill result finaUy from (1.2), (lIA) and (I1.6) for the difl'erent order terms:

110= - ~)

z

<

110

on z= 110

z

<

YJo

(II. 7)

Il 0<])1_

of

+ 2 \7<])0' \7<])1 + rh -:>,0 (\7<])0)2 +

K(l]l O:~O-

⁺

~<])1 î

^{= 0}

uZ (jZ- uZ /

1]1= - _ _'---::::<])~1____;=,____:__

K

+

(o<])%z)

(*)

on z= 1]0 ^(lL8)

on Z = 1]0 (II.9)

\72<])2

=

0 Z

<

1]0

( 02

<])1 + 0<])"\1+ ') \7 \7 + ('\7 )~ ^{+ 0 ('\7 )2 + .) 0 (\7 0 \7 )}

n \ 1]1ozOI

-ar/ -

^{<])0' <])2 V <])1 - 1]2}~z ^V <])0 1]1- oz \ <])0 . oz <])0 + + ') 0 ('\7 '\7) + I{( . 02<])0 + 1]120³<])0 + 02

<])1 0<])2) - 0

j

- 1]1 -:;;;- ^V <])0' ^V<])1 \1]2~ ^{-2C>'l 1]1}~+ --;:;;:- - ,

uZ \ uz- uZ" uZ- uZ

1]₁

(.2h,CJ-=-<JJ.Q.

_{2 OZ2} + 0<])1)_oz +

<])~

_- ^\_,

1]2 = -- K + (o<])%z) -

Obviously, the procedure may be continued for higher tenus.

Equation (lL7) for the <])0 are, as expectable, the exact equations of steady flow.

Equations (IL8) and (IL9) are the linearized equations for tirst and second order tenus, posed on the steady

Z

=

1]0 free surface.

In two important particular cases the equations (I1.8) and (lU)) may be substantially simplitied:

a) When the steady unperturbed state is the uniform flow (Fig. l a).

In this case it is appropriate to transfer the equations(II.7), (11.8) and (II.9) to the axis X, Y, Z of Figure 1.

Hence, b,\' the transformation:

X= x cosa - zsin a Y=y Z = ^X sina + Zcosa ^{(ILl 0)}

(II.12) (ILlI) z

<

0

on Z=O

we obtain for the different approximations:

<])0= KsinaX 1]0 = 0 \72<])1 = 0 n 0<])1 + K sin a 0<])1 + K cos a 0<])1 = 0

of

oX oZ

l <])1 1]1= -

If

cosa

\7 2<])2= 0 Z

<

0

0<]).) + K ' 0<])., + K 0<]).) 0 ( 0<])1 • + 0<])1 '\

Il o f - s m a oX"' cosa

oi'

^{= - Ill]l}

Zif' \ -

-oX - sma oZ cosa ) - 1

-... (_{.. oX )}0<])1

V

+ (_{,0Y )}0<])1

V

+ ( o<])L \ 2_oZ)

J-

_{_} ^{KI] ( _}₁ ^{sin2a 0}_OX2^{2<])1 + cos2a 0}_OZ2^2<])1) _{on Z =} 1]0 (II.13)

( 0<])1'

+

^{0<])1 \}

+

1]1 \ - - - sma ---~cosa 1 <])2

\ oX oZ J

I]~=--- - - -

- K cos a

Steady unperturbed free-surface Surface libre non perturbée, en régIme continu

<'- Moving free - surface Surface libre en mouvement

z

!

^Moving

^~(x,Y,t)

free - surface Surface libre en mouvement

"-Unperturbed free-surface (rest)

Surface libre non perturbée

(ûu repos)

(a) ('11)

A moving fl'ee-surface. The steady unpel'tul'hed

state is: al the unifol'm f1ow. b) the l'est. JI/ Une surface libre en mouvement. Le régime continu non perturbé comporte: a) Ill! écoulement uniforme;

h) l'eau au repos.

(*) The symholic notation has the meaning :

OqJo oqJ1 oqJo oqJ1 o<po 0<P1

\f<p .\f<p

= - - + - - + - - ,

etc.

o ^l OX OX oy oy oz oz

903

G. DAGAN

b) An even more particulaI' case is that of initial l'est, Le. the case (a) with a= 0, (Fig. 1b). It therefore reduces, by making a = () in equations (ILll) (II.l2) and (lI.l3) to:

(jlo=() (II.14)

(I1.16) (lI.15 )

on z

=

'10

z <

0

\72(jl] = 0

n

3

^J(jl]

+

^K~(jl~

=

0

01

oz

on z

=

'110 '1]

= - i{~ \

\7 2(jl2

= ()

z

<

0

n

^0(jl2₀₁

+

¹⁽O(jl~

_oz

^{= _(jl]_}~

(n

~ee.L

+

^K O(jl]) __ [ ( O(jl]V

+(

^O(jl]V

+

^{(O(jl]_\ 2}

J

1(

oz \

⁰¹

oz _ ox) oY ) oZ ) _

1 O(jl] (jl2

'12= 1(2-

oz

^K

Obviously, the steady state is a particular case for which aIl the derivatives in respect to 1vanish.

Some problems of unsteady bidimensional fiow have been solved by the first approximation (equations II.15) by Polubarinova Kotchina (1959), Beliacova (1955), Galin (1959), Meyer (1955), and for steady fiow by Kir- kham (1958) and Dagan (1964).

III. An example of expansion in small parameter of an exact solution The approximation based on expansion on a small parameter exposed previously has a formaI character, unless the convergence of the series (11.1) is proved.

A general proof of the convergence will ensure the validity of the expansion, on one side, and on the other side, may constitute an existence and uniqueness proof for the solution of the original nonlinear problem.

Unfortunately, beyond a simple case studied by Priazinskaia (1955) (by other means than the approximative method discussed here), there are no such proofs for problems of unsteady free-surface fiow in porous media.

In this chapter the converging character of the expansion in a small parameter will be shown in a particular case of an already exactly solved problem.

For this pm'pose let us consider the steady bidimensional fIov>' towards an horizontal array of equidistant sinks of equal strength Q, supplied by a uniform ascendent current at infinity 1. 'Vith the notations of Figure 2 il is Gbvious that Q= 2al.

The qualitative analogy between the free-surface shape and the standing waves is obvious.

The solution reproduced below was derived by van Deemter (see Schilfgaarde, Kirkham and Frevert, [1956]).

Due to symmetry, the fiow is investigated only in the strip SPORS (Fig. 20).

With Z = X

+

^iY(*) a complex variable and

n

= (jl

+

^i\ji the complex potential, the relations:

(IIU) and:

(IlI.2) Z -

+. +.

~

(

^{l '1' - 1_ -} ~

+

^{2 L ln '1'}

+

¹

î

- a le 1 7t Il '1'

+

1

+

~ ¹⁽ '1'

+

¹

+

~

/

map the Z and

n

regiol1s of fiow (Fig. 2a and 2 b) onto an auxiliary '1'= s

+

ir plane (Fig. 2e).

The relations (I11.1) and (lII.2) contain aIl the elements of the solution. K-as ordinary-is the hydraulic conductivity; a, b, c are characteristic dimensions (Fig. 2); ~-is an auxiliaryconstant and is determined, for instance, by the given pressure in a certain point.

From (I1U) and (I11.2) the foIlowing relations result:

7tC

=

ln 2

+

~

+

^{2_l_ln 2}

+

~

a ~ K 2

7tb 2

+

~ l 2

+

^~

-a

^{= ln-~-}

+

^{2 J{ ln](1}

+

~)

!:(c --b)

=

2 l ln (1

+

^Q)

K K

^f'

(lU.3) (IlIA) (I1I.5 ) The purpose of the foIlowing derivations is to demonstrate that

n,

and consequently ^(jl, can be expanded in a convergent series of the powers of a little parameter E, while the consecutive tenns of the expansion no, nI'

n

2 ••• are determined in a rectangular strip in the Z-plane, where the free-,surface is replaced by a horizontal segment (according to equation 11.15).

(*) In this chapter Z will signify a complex variably, Y the ordinate and X the absciss. '1'0 avoid confusion with the notation of the remaining chapters, capitll1 symbols will be used.

904

Z - plane

Plan Z

n -

^plane

Plan 12

p

y' Z'- plane-,olan Zi R'

x'

(d) The steady bidimensional deep flow towards an

array of sinks. /21 L'écoulement permanent plan et profond vers un

ensemble de puits.

Let the strip S'P'O'R'S' be the region of flow, in which the approximate solution will be derived, in the new Z'

=

X' + zY' plane (Fig. 2d). This region is mapped onto the same auxiliary '1' plane (Fig. 2e) by:

Z'= a

+

ib'

+

ⁱ ~ ^{1 '1' - 1 -}

W

'Jt n T+1+W (JII.6)

where b'-the elevation of the straight line replacing the free-surface in the approximate solution-is taken so that W=~. This requires that:

I_CII - a l_{- -} n - - -2+~

'Jt ~ (JII.7)

Now, the images of the points S', pl, 0 ', R', and 5, P, 0, R in the '1' plane will be identical.

The comparison between (JIL3), (JIIA) and (lII.7) show that b

<

b'

<

^c.

Let (l/K) = ^E be taken as a small parameter. The relations CIJI.3), (111.4) and (IJI.7) show that for ^E~0, Z ~Z' and (c - b) ~

°

^i. e. for the vanishing small parameter the flow region degenerates into the non perturbed scheme and the maximum free-surface drop (c - b) tends to zero.

Introducing the new dimensionless variables:

--_{Z =} Z - a - i b'

'Jt - - - -

a

-- n-Kb'

n

= 1: ---'---=-- Ka

by which the origin is transferred to R', the relations (JILl), CIJI.2) and (IJI.6) transform into:

? ë _ • ( '1'- 1 - _~ ') (2

+

~) ('1'

+

^{1) )}

Z - z \ ln '1'+ 1+ ~ ⁺ ~^{Eln 2('1' + 1+} ~) - _ (2 + ~)2(T2-l)

n -

Eln 4 [1~-=Cl +-~)2J-

-, . T+1+~

Z - z ln ----.---.-.. -

T-l-~

which map the new variables on the 'l'-plane. From (111.11) the mapping of '1' on Z' is given by:

'1'= --i (1 + ~) cotan--Z' 2

(111.8a)

(lII.8b) (III.8c)

(lII.9) (JIU0) (JILll)

(JILl 2) From (IIU)), (111.11) and (JILl 2) :

Z -Z' = E2i ln J~_+ ~) ^{[1 - i} Cl + ~) ^{cotan(Zi/2) ] (111.13)} 2Cl + _~) [1 - i cotan CZ,/2)

J

The relation (IILl3) maps the flow region

Z

onto the plain of the approximate solution Z'. It can be written in an abbreviated form as:

- - - - _ . _ - - - _..- - - _ . _ - -- - - -._._...__^._--_..

__

.---_.._---

* N.D.L.R. - Le sUl'lignage appliqué aux symboles Z, Z' et !2 ne signifie pas ici que ce sont des valeurs moyennes, mais que cc sont des variables sans dimensions,

905

G. DAGAN

The function

t

^(Z/) is regular in the strip SlpIOIRIS, its maximum is reached for:

If

^{(Z/) 1}z'=o

= 2

^ln

21 ~

In each point of the ZI plane the equation (111.13) defines a circle of radius

:Z -

^Z/!. The function fi(Z/) can be expanded in Taylor series into such a ch'cle since'TI(Z/) is analytic in the half-strip SIPIOIRISI and takes real values on the upper boundary YI

=

0; hence, it may be analitically continued beyond this boundary and obviously beyond SIRI (Fig. 2d).

Hence, the expansion:

:n

[ZI

+

^{Et(Z/)] =}

n

(Z/)

+

^Ef(Z/)

dE ⁺

^E2

^f

^(Z')2

d~n

^-l-

dZ^I 2 dZ/2

is convergent in each point of the ZI-plane, and according to (111.13) represents the fi(Z).

After some computations, the first two terms look:

(111.14)

2ln 2

+

~

+

n

(Z) = ^{E \} 4 (1

+ _~) _ ( +

^{ln [ 1}

+ _~

⁽²

+ _~)

^cos2

_~]

cp expansion will result from:

1_

^{E2 2}

i~

⁽²

+ _~)

^{sin'lI ln (2}

+ _~)

^[2

+ _~

⁽¹

+

e-^iZ')]

+ ...

\ 1

+

~(2

+

~) cos²(ZI/2) 4Cl

+

~)

. (III.15)

cp(XI,Y') = OlD(Z/) and will be obviously a convergent series in E.

In this case the convergence is assured for arbitrary E, since b, c and b' have been taken according to the exact solution.

IV. Two applications of the second order theory The solution of problems of free-surface flow in porous mediums by second order approximation will he illustrated by two simple examples of bidimensional flow.

a) THE STEADY FREE-SURFACE FLOW THROUGH A POROUS DAM

(lV.l) This case has been studied by Muskat (1937) and Polubarinova-!{otchina (1952) by exact methods.

For the approximate solution, let us take E= (llH/L) as a small parameter (Fig. 3a).

cp and 11 are expanded up to third order tenns as following:

cp

=

ECP1

+

^E2CP2

11

=

^E111

+

^E2112

since the non-perturbed state corresponding to E

=

0 (ll

=

0) is the l'est.

The boundary conditions for cp, CPl and CP2 are chosen according to (1.2), (11.15) and (H.16) (*) and given in Figures 3a, 3 band 3c.

---

y y

Free - surface (EQ. 12) Surface libre

l

~

l <l

ay =0

V1:p=O

^:::s:::

^a

H

^<l_:::s::: H-~H _Il¹ ₁

V2.cA=O

1 ^{Il H-~H}

a

_Il

a

_H-~H

1

-e:' _~

^Il

Il

Q.P-O ~1=0 4

B- ay-

^w

L L

L

(a) (h) (c)

Seepage through a dam; houndary conditions for:

a) cp, b) <Pl, c) CP2. /3/ Percolation par un barrage; conditions cm.T limites correspondant ci : a) <p, h) CPI, c) <P2.

(*) Note that Y replaces Z.

906

The solution for CfJl is quite simple:

CfJl = - IŒH [1 - (X/L) ] = _ KL [1 - (X/L) ]

E (IV.3)

. 7t(2n

+

^1)X

sm

L

1)1 - L Cl-X/L)

and represents the uniform flow through the confined rectangle of Figure 3b.

For CfJz (Fig. 3c) a Fourier series representation is suitable:

;. A h n7t^(Y

+

H - .6..H) . n7tX CfJz = "" n cos SIll- L

1

L

CfJz, in this form, satisfies Laplace equation and aIl boundary conditions excepting on Y = O.

On this boundary:

~;z

^{= -}

I~ (~k1 Y

^{= - K}

Computing An from (IV.5), it is finally obtained:

"" co h 7t (2 n + 1) (Y+ H - .6..H) " 7t (2 n + 1) X

4 KL s , L sm L

CfJz=- -~- ~ , . .

7t- "'"-J (2 n + I)Z sinh 7t⁽²ⁿ + 1) (H - .6..H)

o

L

t 1 7t (2 n + 1) (H - .6..H) CfJ~ \ 4L,,~ co an1 L

1)z= - Ii y=o=~

2..J

(2n

+

^I)Z

o

Consequently, by (IV.l) the final expression of 1) is:

t ^"·1 7t (2 n + 1) (H - .6..H) . _7t---'-(2_n--d+---'1)_X_

"" co an1 . sIn

1) (X) = ( 1 _ X') .6..H + ~(ÂH)2 " L . L

L \ L L 7tz\ L ~ (2 n

+

^I)Z

o

(IV.4)

(IV.5)

(IV.6)

(IV.7)

(IV.8)

The exact expression for the flux through a vertical sectio~iisgiven by:

_ (n-!m+^1J OCfJ

Q -

Jo oX

dY

The evaluation of the discharge

nV.9) By the expansions (IV.l) it becomes:

(n-SJ:I+e^1J1+e^21J

o(

^{':>m ':>m )} (n-!>H':>m [':>m 1 (n-!>H~tll ^]

Q=

Jo -\

E

~X +

^EZ

~;t

^{dY = E}

Jo _~X1

dY

+

^{EZ_ 1)1}

~x1

^Y=O

+ Jo _~i:

dY _

+

^0(E3)

(IV.I0) From the solutions of CfJl and CfJz (lV.3) and (IV.6) the discharge, up to third order tenns, resulting from (IV.I0) is:

Q= EK (H _ .6..H)

+

E2 KL = K HZ - (H - .6..H)2

2 2L

However, Qfrom (IV.ll) is the exact value of the discharge (Polubarinova-Kotchina, 1952).

This striking fact suggests that the higher terms in Q expansion cancel.

(IV.ll)

Comparison with Dupait solution In the particular case in which not only ÂH/L is smaIl, but H/L also (shallow depth flow) the solution (lV.6) for CfJz may be replaced by the moresimplê one derived by Polubarinova-Kotchina Cl952) for an infinite long strip:

CfJ2 = 2

CH ~.6..H)

^{[( X -}

~ y-

^(Y

⁺

^H

~.ÂH)Z

^]

+

^{const (IV.12)}

This solution cannot satisfy the condition CfJz = 0 at X = 0 and X = L, but is able to represent fairly CfJz in the central portion of the strip (X= Lj2) .. With CfJz = 0 at X= 0, L and Y - 0, (IV.12) becomes:

2 K H) [X (X - L) - Y (Y

+

2H.,-- 2 ÂH) ]

( H - Â

The solution based on Dupuitassumption for the seepage through a dam is:

HZ-(H-.6..H

+

^{1)2 _ 2Qx}

K

(IV.13)

(IV.14) 907

G. DAGAN

By (IV.ll) and S0l11e transformations OV.14) becomes:

1]

=

ÂH - H

+

^H

1-

^{1 - ÂH}

(2 --

^ÂH

^'1

~J-¹¹²

_ H H ) L

Supposing ÂH/H small, the square root l11ay be expanded into a series and (lV.15) yields:

1]/L= ÂH (1 __ X)

+ 1-

^{(ÂH)2 X}

(1 _

^X)

⁺ 0

^(ES)

L L; 2 L H L

Hence, the tenn in E2 by the linearized solution for 1]/L is according to (lV.13):

l

^{X - X}

1]2/L

=-~

_{K y=() 2 (H - ÂH)} ^{( 1 - -}_L

and the sil11ilar onë frol11 the solution based on Dupuit assul11ption according to (lV.16) IS:

(lV.15i

(lV.16)

(lV.17 )

. X(1 X)

1]2 (Dupmt)/L - 2H

- L

'l'12/L from (lV.17) and (IV.l8) are very closed if (IlH/H)

«

1.

In conclusion, for:

(IV.18)

llH ÂH

L «

1, H/L

«

1 and

I f « ],

the linearized theory and the non-linear one based on Dupuit assumptions give close results up to the third order terms of the free-surface equation.

Deep t10w throllgh a dam

'

''TI H 1 ^'lt(2Il

+

1) (H --IlH)

1\ len I:;-

> ,

the cotanh L --- in equation (IV.8) is very close to 1.

flo\\' the free-surface shape is no more influenced by the dam height, and (IV.8) becol11es:

Hence, for deep

oc

_YlSX) = ( 1 _ ~\ ÂH

+

^(IlH)2

--±-

~_sin ^['lt(2Il

+

^{1) X/LI}

L \ L ) L ,L ^'lt2L.J (2Il

+

¹⁾²

o

(lV.19)

1] has been computed from this rapid convergent series up to ten tenns by aid of a high-speed electronic com- puter for llH/L= 0.1. The results are given in Figure 4.

b) DAMPING OF AN INITIAL COSINUSOIDAL FREE-SURFACE

Let us suppose that at t - 0 the free-surface bounding a region of infinite depth (Fig. 5) has the shape:

1]= ECOSo:X

0,4

77

L 77,

0,2 L

51 An initial cosinusoidal profile of the free-surface.

Profil cosinzzsoïdal initial de la surface libre.

= - KE COSCXX

cp

(X,O,O)

=-

K y

r;

(X,O)

41 Seepage through a dam; free-surface profile in the case of deep flow (H/L>l).

Percolation par un barrage; profil à surface libre dans le

cas d'un écoulement profond (H/L> 1).

0,8

X 772

L

---c-

0,1 0,198 0,2 0,299 0,3 0,362 0,4 0,396 0,5 0,407

0,4 0,6 0,2

(IV.20) at t = 0

Where E= Aa, A being a length determining the amplitude of 11(*).

Up to third order tenus, CfJ and 11 have the expansions (IV.l). According to (IV.15) CfJ1 satisfies the conditions:

n

0;/ +

^K

_~~

^{= 0 on Y= 0}

CfJ

= -

K¹¹¹

= -

KcosaX on Y= 0

The harmonie function, vanishing at - co, satisfying (IV.20) is:

CfJ1= - Keœ[Y- (Kln)t]cos aX

and:

111= e-œ(Kln)tcosaX

The second approximation fulfills (II.IB) which becomes in the present case:

(IV.21)

and:

112

=

^a cos²aXe- 2œ (Kln)t-~ = 0 K

on Y=O

on Y=O at t⁼ 0

(IV.22)

The appropriate solution for CfJ2 is:

CfJ2

=

^aK e-2œ(K2In)t(1

+

^{e 2œYcos2aX)}

2 (IV.23)

Hence, from (IV.I), (II.15), (IV.21) and OV.23), the solutions of CfJ and are:

CfJ

= -

Keœ[Y- (Kln)t]cosa X

+

^E2 aK_ e - 2œ (Kln)t('1

+

^e2œYcos2aX)

+

^0CE3)

2 (IV.24)

11= ^Ee-œ(Kln)tcosaX

+

^0(E3)

The striking fact shO\vu up in (IV.24) is that the free-surface profile is approxil11ated by the first term at least up to third order.

v.

Conclusions Itseel11S that the Iinearized theOl'y or different orders makes possible the solution of a large cIass of free-surface flow problems. The great advantage of the approach derives from the possibility of using the great variety of l11athematical solutions for partial derivatives equations with Iinear boundary conditions. It is apparent that a substantial progress may be achevied in two distinct directions: the thoroughly l11athematical study of the convergence of the series expansions and the derivation of newl solutions for different particular cases.

List ofSymbols

a, b, c, b ' - H .6.H

-

k -

K l - L

-

Q a -

~,

W

-

y -

geol11etrical dimensions;

height;

level difference;

intrinsic hydraulic conductivity;

hydraulic conductivity;

specifie discharge (LIT);

length;

discharge (L2/T);

angle;

paral11eters of Integration;

specifie weight;

CfJ - velocity potential;

OCfJ _ normal derivative;

on -

E - smaIl parameter;

IL viscosity (dynamic);

n

free-surface elevation;

il complex velocity potential;

tjJ stream funetion;

p density;

o

^{(E2) _} power series in E starting with E2•

(*) The general case for a finite-depth region of flow was solved by Meyer (1955) by the first approximation.

G. DAGAN

References

BEAR (J.) and DAGAN (G.). - The use of the hodograph me- thod for ground water investigations, Technion-Israel Instiiut of Technologu, Hudraulic Laboratory, P.N. 24 (1962) .

BELIACOVA (V.K.). - Unsteady flow of groundwater towards an horizontal drain, Prikladnaia Maiematika i Meha- Ilika, vol. 19 (1955).

DAGAN (G.). - Spacing of drains by an approximate me- thod, Proc. A.S.C.E., .TOl/m. of Dr. and Ir. Div., vol 90, No. 1 (1964).

GALIN (L. A.). - Unsteady seepage of groundwater towards a narrow ditch, Prikladnaia Maiematika i MehaniIw, vol. 23 (1959).

I{IRKHA)[ (D.). - Seepage of steady rainfall through soil into drains, Tl'ans. Am. Geoph. Union, vol. 39, No. 5 (1958) .

MEYER (R.). -- Quelques résultats théoriques récents con- cernant les écoulements des nappes d'eau souterraines, La Houille Blanche, No. 1 (1955), No. 5 (1955), No. 1 (1956) .

l\IUSKAT (M.). - The flo'w of homogeneous fluids through porous media, McGraw-Hill, New York (1937).

POLUBARlNOVA-I{OTCHINA (P. Ya.). - Theory of ground water movement,Governmenial Publishing Office for Technical and Theoretical Literaiure, Moscow, (1952), (also Prin- ceton 1962).

POLUBARINOVA-I{OTCHINA (P. Ya.). - Ground water move- ments at water level fluctuations in a reservoir with a vertical houndary, Prikiadnaia Maiematika i Meha- nika, vol. 23, No. 3 (1959).

PRIAZINSKAYA (V. O.). -- The prohlem of plane unsteady motion of ground waters, Prikiadnaia Maiemaiaw i Me- hanika, vol. 23, No. 5 (1955).

SCHlLFGAARDE (J.), I{IRKHAM (D.) and FREVERT (R.1{.). - Physical and mathematical theories of tHe and ditch drainage and their usefulness iu design, Iowa, Research Bulletin, No. 436 (1956).

Résumé

Théorie linéarisée, du deuxième ordre,

de l'écoulement à surface libre dans les milieux poreux par G. Dagan ':'

On examine l'écoulement à surface libre d'un liquide incompressible et homogène dans un milieu poreux isotrope, homogène et indéformable.

Pour résoudre exactement un tel problème, il est nécessaire de déterminer le potentiel de vitesse, <p, comme fonction harmonique (équation 1.1) respectant une condition aux limites non linéaire (équation 1.2) sur la surface libre inconnue en mouvement (équation 1.3).

Une méthode approchée consiste à développer les expressions, correspondant au potentiel, et à la surface libre, en Une série exponentielle d'un paramètre E de faible valeur (équations 11.1, Il.2), semblable à l'approxi- mation des ondes infinitésimales. On dérive ensuite les équations définissant les deux premiers termes (équations ILS, 11.9). Alors que les fonctions restent harmoniques, les conditions à la surface libre deviennent linéaires, et sont posées pour une surface permanente, et a priori connue. Deux cas sont examinés en détail, d'une part, celui d'une surface permanente correspondant à un plan incliné (équations 11.10-11.13, fig. 1a), et d'autre part, celui d'une surface plane et horizontale (équations Il.14-Il.16, fig. 1b).

On examine la convergence des développements en fonction d'un paramètre de faible valeur, pour le cas d'un écoulement plan se dirigeant vers un ensemble de puits (fig. 2), pour lequel Van Deemter avait trouvé une solution exacte. Il apparaît la possibilité de développer le potentiel complexe il en une série (équation UL15), conver-

gente pour une valeur arbitraire de E. -

Deux exemples de solutions du deuxième ordre sont présentés.

Dans le eas d'un écoulement plan permanent par un barrage, le profil de la surface libre est défini par l'équa- tion IV.S. On montre que l'on aboutit à une évaluation exacte du débit, et que ces résultats sont voisins de celui obtenu à l'aide des hypothèses de Dupuit, à condition que l'écoulement soit peu profond (équations IV.17, IV.1S).

On calcule le profil de la surface libre pour le cas d'un écoulement profond (équation VL19, fig. 4).

L'amortissement d'une surface libre initiale de profil cosinusoïdal est un exemple d'un écoulement non permanent. La solution est exprimée par l'équation IV.24. On croit pouvoir obtenir une importante classe de nouvelles solutions à l'aide de la théorie linéarisée.

-~---_._---_.---_._-_._-~-

• Lecturer, Hydraulic Lahoratory, Technioll-Israel Institllte of Technology.