Application of a new variational technique for the study of seepage as a free boundary problem

0895-7177/90 $3.00+0.00 Pergamon Pressplc

Math1Comput.Modelling, Vol.14,pp.781-784,1990 Printed in Great Britain

APPLICATION OF A NEW VARIATIONAL TECHNIQUE SEEPAGE AS A FREE BOUNDARY PROBLEM

S. M.

Makky

and

A.

FOR

THE

STUDY

OF

Ibrahim

Mathematics Department, College of Science, P.O.Box 41005, Al-Jadria, Bagdad, IRAQ

University

of Bagdad

Abstractl Seepage in soil is formulated as a free boundary problem. An equivalent variational problem is formulated, then an approximate solution for the variational problem is sought. By representing the velocity potential and the unknown boundary as linear combinations of certain classes of functions (generalized Ritz method) the variational problem is recast to the realm of mathematical programming. A computer programme is developed and used successfully to solve the seepage problem from a triangular channel to underground water beneath the channel. The same procedure is utilized to solve the seepage problem in earth dams. The results seem to be very realistic. Keywords

Seepage; free Ritz method

boundary

problems;

variational

methods;

INTRODUCTION The above method is used successfully to solve problem of seepage from a the through homogeneous, triangular ditch isotorpic soil to underground water below The ditch is assumed so long the ditch. that the seepage can be considered two dimensional and in the vertical plane. A computer programme is developed and writtten in BASIC for an HP9845 minicomputer and implemented for the study of The results are the seepage problem. given in graphical form.

Seepage problems play a vital role in many aspects of life; in agriculture, irrigation, design of dams & foundations, in water & soil polution, to name but a In few. most seepage problems the surface separating the saturated soil its from surroundings is unknown beforehand. As a matter of fact determining the seperating surface constitutes a major part of the solution. Thus, most seepage probleas can be formulated as free boundary problems. Many investigators have tackled free boundary problems, and a variety of methods are developed for this purpose l~l,[zl. In this article a recently developed variational approach for free boundary problems is adapted to solve a particular seepage problem. The method can be summarized as follows. An equivalent extremum variational problem is sought whose critical point is a solution for the free boundary problem. Then the function representing the unknown boundary as well as the other variables of the problem are expressed AS linear combinations of the elements of preassigned complete sequences of functions (generalized Rits method). -fhis means that the problem is recasL to the realm of mathematical programming.

STATEMENT

OF THE

PROBLEM

The problem is to study the seepage from a ditch to an underground water beneath it. It is assumed that: lThe cross section of the the level isoscele triangle; uid in the ditch is constant. 2due and

ditch is an of the liq-

The ditch is so long that the flow to seepage can be considered steady two dimensional.

The soil between the ditch and the 3underneath underground water is permeable, homogeneous and isotropic. 781

Proc. 7th IN. Conj on Mathematical and Computer Modellin~

782

The region below the ditch which is eaturated by water is called the flow region. saturated boundary between and The unsaturated regions (neglecting capillary forces) constitutes one of the unknowns of the problem.

h(0) h'(O)= h(E)

0

(9c)

-m

(9d)

h'(E)=

V

Mathematical

Vn(h(y),y)

of

the

Problem

Relative to the coordinate system depicmodel corted in Pig.1, the mathematical problem described in responding to the the previous section reads as follows [s] VXX + vyy = 0 in the region

(1) R bounded

by OABD.

=o,

for

otxtm

(2)

Vx(0.r)

=o,

for

O
(3)

V(x,c(x))

= E

for

o
(4)

V

(x,0)

V(x,g(x))

,

(9b)

= xo

The reason for studying such a problem is to determine the rate of flow of liquid from the ditch to the underground water for pollution prevention purposes and for agricultural irrigation. Model

(9a)

= xi

Consequently, conditions take the forms: for

0 < y < E

(6;

= 0 ,

for

0 < y < E

(6')

Equivalent

Variational

It can be shown that the variational problem equivalent to the free boundary problem given by differential equation conditions (2-6) reads (1) and boundary as follows: Find the functions xi that render the be minimum.

x < xi

(6)

I(V,h,w)

=

f"" 0

,

0

for

XQ<

x < xl

V and h and the point following integral to

It can be easily must satisfy the

curve

seen that following

dx

dy

+

0

('/x2+ vy')

cCcY

dx dy

(10)

meets here V satisfies conditions satisfies conditions (9).

y = c(x) = mx + L ( straight line is the equation of the representing the edge of the ditch. Y = g(x) , is the equation of the ting the free surface.

(Vx2+Vy2)

f

(6)

bove relations: In the V i.s the velocity potential. the point where the free surface X the underground water.

represen-

the function g conditions: (7a)

SOLUTION

OF

THE

(2-6)

and

A Functio_n__ SatisfxtiKJ&_Non-homoxeneoue _-&u&d&.ry Ce_ndition_s R to the region of flow Devide subregions as follows (see figure 2)

I

four

(Ilab

O
; O
(7b)

KZ = r(x,Y):

o
; L
= 0

(7c)

&I = I(x,y):

x+
; O
I

(llc)

(Xi ) = --‘t.

(7d)

HA = [(x,y):

xq
; L
1

(lid)

g'(x0) g(w)

= L + mxc = -1

the Due to more seems free surface

= E

/m

singularity benoficiaJ

= IZ-I(y)

Thus

conditions

by:

R at. xl jt the represent

Define WI =

by:

x = h(y) the

of t0

(8) (7)

will

be

replaced

h

PROBLEM

RI = I(x,y):

g(xg)

g'

Problem

C(X) x0<

= g(x),for

Vn(x,g(x))=

(6) will

= Y ,

(h(y),y)

The

(5) and

the

following

( 2E y/L

1

functions

1 - ( E yz/Lz )

(12a) (12b)

W2 = E w3 = Wl(X,Y)

(lib)

+ lu[ h(y)

Wl(X,Y)J - xo

I2

(x-xoJ2/ (12c)

Proc. 7th Int. Conf. on Mathematical w4

Wz(x,y)

=

+

[y-

WZ(X,Y)l

Furthermore W = Wi

h(y)

- xc

(12d)

1

let for

on Ri,

i = 1,2,3

6 4

(13)

Note that the above definitions for W renders the functions W, Wx, and Wy to be Moreover the function W continuous on R. Define a function P as satisfies (2-6). follows :

P(x,y)=

x2y

Ly-C(x)lIx-h(y)1

N

c

c Amn

m-0

x"y"

n--O

(14) This function satisfies the homogenous boundary conditions associated with (2-6) for any choice of parameters Amn. Define V by V(x,y)

= P(x,y)+Wi

,for

(x,y)

in

Ri

(15)

It follows that V will satisfy conditions (2-S) for any choice of parameters Amn. The

Form

of

the

Free

Surfac.e

When the free surface is expressed by relations (8) it must satisfy conditions (8a-8d). It can be checked that the function defined by ,y) = By'+

h(xl,B,B

xi + y21BE2+

(m/E)

+

LJ 3(x3-x1)/

E2) - y3[2BE

+

BLJ y"

In

m/E')

OF THE

PROBLEM

To simplify the computational efforts so that a minicomputer (HP 9845) can handle such a big problem. only B, AQO, MO, Aoi are retained. Thus, the number of variables is five, namely xi, B, and A,C, D for k,o, ho, AM. Differentiation is carried by hand, before integration. Gaussian quadrature is used for numerical integration [41. The measurements are taken to be L=l;

M

783

Modelling

SOLUTION

(x-xo)2/

2 [

and Computer

E = 1.5

;

m = 0.5

;

x*3 = 1

Figure 3 shows successive approximations for the free surface, and the curve to which they converge to. REMARKS The convergence of the above described procedure is ensured by the convergence of the Ritz method for variational problems, and by the convergence of the algorithm used to solve the resulting nonlinear programming problem. The convergence of the method of steepest descent is discussed in [51. The convergence of the Ritz method is proved in 16) when the problem has fixed boundaries. The proof is generalized in [?I to cover free boundary problems. The same procedure is used successfully to study seepage through earth dams [a), and water coning in oil wells lo].

+ (2/E3)1 REFERENCES

LZZ

(

Y-E

(16)

5’2

111 satisfies

the

above

ment i oned

conditions. 121

When the form (15) for V and (16) for h are used in connection with the expression (10) for I and when integration is carried out it follows that I becomes a function of variables Ixl,A~<~-,H.B~I. In is words the variational problem other now equivalent to t.he following unconstreined non-linear programming problem: Find the minimum of the function I(Xl,A:r.-,B,BIJI.

There are some efficicnl algorithims to tical solve the above mathemaproblem. l'he mrLhod of programming this steepest descent. is chosc~n t’O1 purpose.

Crank, J. (19841, "Free and Moving Boundary Problems".ClayeDdon P&X.%.%, Ockendon, J.H. and W.R. Hedgkins (Edsl (1975)," Moving Boundary Problems in Heat Flow and Diffusion II, Clere_ndon_press,

Voroglu,

E. and W.D.L. Finn, (1982), "A Variable Domajn Finite Element Analysis of Seepage From a Ditch", in Ghallghar,H.H. and J.T. Oden, fEdsI, "Finite Elements in Fluids", Wiley Interscience Pub. Krylov, V.I., 141 (1962), "Approximate Calculation of Integrals", Tbe~_ __Macmillan Co. Lucnbcrger, D.G. 51 ( 1965 ) ( "Introductlon to Linear and Non-linear PrOgramming", Addj son Wes_l_ey. Kantwvich, 61 L.V. and V.l. Krylov, f 195x I )” Approximate Methods of High131

PI- Anaiysis",

Noordhot'f

Proc.

784 [7]

[8J

7th Int. Conf. on Mathematical

Ibrahim, A. (1988), w On Seepage Problem", H.Sc. Theses, Math. Dept., Science, University of College of Baghdad, (in Arabic) Houssain I. (1987) "Extremum Variational Principles for Boundary value Problems", H.Sc. Theses, Math. Dept.

FIG.(l)

FIG.(2)

SUB31V:S!3\5

3F

Ti(E

FLOW

*SHAPE

REB’ON

AND

[Sl

DIMENSIONS

and Computer

MoaWing

College of Science, University of Baghdad, (in Arabic) Hakky, S.M. and A. Hamad, (19891, "Water Coning in Oil Wells", Proc. of Int. Conf. on Approximation,Optmizetion Computation, Dalian, and China.

OF

THE

DITCH.