strack 1976 a single-potential solution for regional interface problems in coastal aquifers
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8/10/2019 Strack 1976 A Single-Potential Solution for Regional Interface Problems in Coastal Aquifers
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oL.
12. NO. 6
WATER
RESOURCES
Ixrnopucttox
An
important
class
of
groundwater
flow
problems involves
confined, unconfined,
or interface flow in homo-
isotropic
coastal aquilers. Although
well-developed
exist for
problems
ofthis class, these have not
been
applicable
to
three-dimensional
problems
with two
or
of
these
flow
types in
a single coastal aquifer. However,
rost
problems
ol
practical
importance involve
three-dimen-
flow
which
olten can be
treated
in
two dimensions by
flow in
the
vertical
direction.
The Dupuit-Forchheimer
assumption, which
dates back to
end of the
previous
century
[Dupuit,
1863;
Forchheimer,
i86],
has
led
to
simple
approximate
techniques
lor
solving
state
confined and uncontined
flow
problems.
This
as-
which
states that
equipotentials
can be approxi-
by
vertical
surfaces,
often appears
to
be sufficiently
I'or example,
the
cases
of
steady
state confined and
flow may
be handled in this fashion.
A
lormula
lrom
the investigations
of Badon Ghyben
[1888]
and
[901]
combined
with
the Dupuit-Forchheimer
as-
has led to
a treatment
of
interface
problems
in
aquifers
fcf
.
Bear,
1972]
similar to that lor
problems
of
flow.
The techniques nrentioned
above
for problems
of
confined
unconfined flow
and confined and unconlined
interface
involve
some
additional approximations
that are men-
for
the sake
of
completeness.
For
unconfined
flow
actual transition
zone between
water
and air is approxi-
by a
groundwater
table,
and
an impervious
base forms
lower boundary
of the aquifer. For
unconfined and
con-
interlace flow
the
upper boundaries
are a free
water
table
a horizontal
impervious
stratum, respectively.
For both
Copyright @ 1976 by the American Geophysical
Union.
RESEARCH
DECEMBER
1976
A Single-Potential
Solution for Regional Interface
Problems
in
Coastal
Aquifers
O. D. L. Srna,cr
Departnrcnt
of Ciail
and
lIineral Engineering,
nioersity
of
Minnesota,
Minneapolis,
Minnesota
55455
An analytic technique
for solving three-dimensional interface problems
in
coastal aquifers is
presented
in
this paper.
Restriction is
made to cases of steady
state
flow with
homogeneous isotropic
permeability
where
the
vertical
flow
rates can
be
neglected
in relation to the horizontal
ones
(the
Dupuit-F'orchheimer
assumptron). The
aquifer is divided
into
zones defined
by the type olflow occurring. These
types offlow
may
be either confined,
unconfined, confined interface,
or unconfined interface flow,
where
the
interfaces
separate freshwater
from salt
water
at
rest.
The
technique is based upon
the use ofa single
potential
which
is defined
throughout all zones
of
the aquifer.
This
potential
in
each zone can be represented
in
a
way
similar
to
that
suggested by
Girinskii
in
1946 and 1947. The
potential
introduced in this
paper
is
single
valued
and continuous throughout
the multiple-zone aquifer,
and its application does not require that
the
boundaries
between the
zones
be known in advance.
The technique thus
avoids
the
difficulties that result
lronr the discontinuity
of
both the velocity
gradients
and the Girinskii
potentials
at
the boundaries
between
the
zones
and
from
the
unknown
locations
of
these
boundaries.
The
use
of
the
single-valued
potential
is illustrated in
this
paper
for
an analytic
technique,
but it may
be used
with
some
advantage
in
numerical
methods such as finite
difference or finite element
techniques. Applications
discussed
in
this
paper
involve two interface flow
problems
in
a shallow coastal
aquifer
with
a fully
penetrating
well.
The
first
problem
is
one
ol
unconfined interface
flow
where
the upper boundary is
a
lree water
table. The
second is
one of confined interface flow
where
the upper
boundary is horizontal and
impervious.
Each
problem
involves
two
zones.
One zone is adjacent
to the coast and
is
bounded below
by an
interlace
between
freshwater
and salt
water
at rest.
The other zone is bounded
below by an impervious bottom. It is
shown that saltwater intrusion
in the
well
occurs
when
the discharge
of the
well
surpasses a
certain
value
for which
the interface
becomes unstable. The conditions
that
must
be met to
prevent
such saltwater
intrusion are
established for each
problem
and
are
represented
graphically.
types of
interface
flow the
salt and
fresh
groundwater
are
assumed
to
be
separated
by
an
interface rather
than
by
a
transition
zone.
Furthermore.
the flow rates
in the saltwater
region are
assumed to be negligible in relation
to the flow rates
in the freshwater region.
L.ach
of the four flow
types mentioned above
(confined,
unconfined, confined interflace,
and unconfined interface) is
usually associated
with
its
own method of solution. The
use of
the
potential
lunctions
introduced by
Girinskii
in
1946
and
1947
for
confined, unconfined,
and unconfined interface flow
reduces the mathematical
differences
between these three types
of
problems
to
differences in the
expressions of the
potential
in
terms ol
the
head.
The advantage
of the
use
of these
potentials
is that the
potential
as
a
lunction of
spatial coordinates
is the
same for
problems
of different flow
types and the
same
bound-
ary conditions.
Girinskii's
11946,
19471
potentials
are capable of
incor-
porating
a
permeability
that
varies in
the
vertical
direction,
an aspect
that
will
not be explored further in
this
paper,
where
the
permeability
is
taken to be a constant. The
Girinskii
poten-
tials for
confined and unconfined flow
can
be
applied to mixed
confined-unconfined
aquifers and
together represent a func-
tion that is single valued
and
continuous
throughout
the flow
region
[cl.
Arauin
and Nunterou, 1965,
pp.
291-296).
The
po-
tential introduced
by Girinskii
U947)
for unconfined interface
flow, however,
does not
combine
with
the
potentials
for
con-
fined
and
unconfined flow to produce
a
single-valued
function.
Various
authors
have
studied
two-dimensional
problems
in
coastal
aquilers.
Exact analytic
solutions
lor
two-dimensional
flow
in the
vertical plane
of deep aquifers involving horizontal
drains
were presented,
for
instance,
by
Ackermann
and Chang
ll97ll,
Bear
and Dagan
l96al,
De Josselin
de
Jong
11965l,
and
Strack
U972,
1973). The
influence of the
drain on the
form
and position
of
the
interface,
being of considerable
practical
I
165
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I
166
Srnecr: Coesut AeurFERs
Casewherea:0
mportance,
was
determined
by these authors. It
was
shown
by
Bear
and Dagan
|964a1
and
Strack
[1972,
1973)
that
in-
stability
of the interface,
leading
to saltwater intrusion in
the
drain,
occurs
il
the discharge
of the
drain
reaches
a certain
value.
Bear
and Dagan
[964]
and
Henry
ll959l
presented
exact solutions
for
problems
of two-dimensional
interlace flow
without
wells
or
drains
in
shallow coastal aquifers where
the
lower
boundary
is formed
partly
by an interface
and
partly
by
a horizontal
impervious
stratum.
Combinations
of
vertical wells
with
a shallow
coastal aqui-
fer lead
to three-dimensional problems.
A simple
method of
solving
such
problerns
is
presented
in
this
paper.
The
solution
is
described in
terms
of
a
single-valued
potential
which
is
continuous
and harmonic
throughout
the aquifer. In zones
of
confined, unconflned,
and unconfined interface
flow the
poten-
tial is equal
to the
appropriate
Girinskii
potential
plus
a con-
stant,
determined from
the
condition that the
potential
be
single valued
throughout
the aquifer.
Girinskii did not
define
a
potential
lor
zones
ol
conflned interface
flow,
and
a
new
expression
for
the potential
applicable
to
such zones
is
pre-
sented.
DrscuencE Vrcron
AND
DISCHARce PorrNueL
The
discharge vector is
defined as
pointing
in
the direction
olflow
and having
a magnitude equal
to the discharge flowing
through
a surface
perpendicular
to the direction
of
flow,
of
unit width
and of a height
equal to the thickness
of the flow
region.
According
to
this definition the components
ol the
discharge vector
are found
by the multiplication
of the cor-
responding
components
of the specific
discharge
vector
by the
thickness
of the
flow region.
If
the vertical component
of
the
specific discharge
vector
is
neglected,
according
to the Dupuit-Forchheimer
assumption,
and if
q,
and
q,
represent the
two horizontal
components ol
the specific
discharge vector, Darcy's
law lor a homogeneous
isotropic
aquifer reads
where
it is
noted
that k is a
constant.
The
expres.
-::.
U/(2a)1k(a
*
)'
and
k
are,
by
definition,
potentia.s
:f
the
discharge
vector
for
cases
where
a
is unequal
to zert'
,:u
equal to zero, respectively.
For the cases of unconfined
-,
o
confined
flow,
and unconfined interface flow these
pote::
-l
are special
cases of functions introduced
by Girinskii
t'.-t
1947],
which
are valid for
the
general
case that k is a
ful:-.
:.:
of the
vertical
coordinate. Arauin
and
Numerou
[1953.
.;'
discussed these functions, referring
to them as Girinskii
p,:::'r-
tials,
for
cases
of
conflned
and unconfined
flow.
By delining
the
potential
as
Casewherea
f
0
A:ka(+13/a),+C
Casewherea=0
=ka+c
:
where
C stands
for
a
constant,
it
is
seen
from
(4)
th.:
-:r
discharge
vector
equals minus
the
gradient
of
,D;
i.e.,
Q,:
-i:/ax
Qy:
-a/ay
The
difference between Girinskii
potentials
for homoge:=:
-.ti
permeability
and
the
lunction
iD
is
the occurrence
of the
::--
stant C. It
will
be
shown
below that introduction of s-::
.
constant
is
a necessary condition for
single-valuedness
;:
-::c
potential when
one
is
dealing
with
cases
in
which
dii:-::^
types
of
flow
occur
in
the
same
aquifer.
The
governing
differential equation for the
potentltr. i h
obtained from the
continuity equation
in
terms of the
:---
ponents
of the discharge vector, which
is
Q,
- -A--
t
-
;-
='
where
1 represents
some constant influx into
the aquier
'-:"r:
either above or below. The
differential equation in tern:.
,
r
is readily
obtained from
(6)
and
(7)
to be
az
+2
__r
x2
'
y'
DrscnrprroN
or e SHer-low
Coe.srel
Aqurren
Problems of interface
flow often occur in coastal aq-
'r-
where
fresh
groundwater
flows
from land to coast abo\3
irrl
groundwater
that is in
direct connection
with
the sea. A :r:
:-,r
unconfined
coastal
aquifer
is represented in Figure
1.
,i::':
the
upper and
lower
boundaries of
the
flow
region are
lo:-..r
by
a
phreatic
surface
(a
possible
capillary zone is
negle;-::
and an
interface,
respectively.
The treatment
of
problems
of interface flow discussed
::
-:i
paper
is
based upon the lollowing
simplifying
ssflpi---,:
mentioned
in
the
introduction.
l. The flow
rates in the saltwater zone are small in;:::-
parison
with
those in
the
freshwater
zone and can be
negle;::
2. An interface rather
than
a transition zone sepa::=
lresh
groundwater
from
salt
groundwater.
3. The flow rates in
the
vertical
direction are
negligi:.:
ri
relation to those in
the horizontal direction. The head
..::I
any one vertical
can then be taken to be constant and eq;.
.r
the
head
at the
point
of the
phreatic
surface on that
ver:.-:
e,:
-
fiw*ot
e"
=
-
fiwrot
,a ,
,=-K
ax
qn=-k
av
(l
)
where
k is the
permeability,
6
is the head, and
x and
y
are
Cartesian
coordinates in
the horizontal
plane. (The
specific
discharge vector points
in the direction
of flow
and has a
magnitude
equal to the
discharge flowing
through
a
unit
area.)
Representing
the
components
o[
the
discharge
vector
as
Q,
and
Qn
and the
thickness
of the aquifer
as ,, one obtains
Q,:
hQ,:
-0,
*
Restriction will
be made in
this
paper
to
types
of
flow
where
the
thickness
of
the aquifer may
be
represented
as a linear
function
of
the
head:
h:a+
(3)
Examples
of such
types offlow
are unconfined flow, where
the
head
@
is
equal to the
elevation
of the
phreatic
surface above
the impervious
base
,
and
confined flow,
where ft
is
equal
to
a
constant, H.
In the former
case,
constants a and
p
are one
and
zero,
respectively,
and in
the latter
case,
a
is
zero,
and
p
is 11.
Substitution
of
(3)
for
h in
(2)
yields,
in considering
sepa-
rately
the case
where
a equals zero,
Casewhereaf0
Q":
hQn:
*oo
#
(2)
*Woar*l
filia"r.
l
,
--
Qn:
-
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Stnecr:
Coesrll
AeutFERs
li
intuface
6lt
lvtlA,wfer
^t
lebl
PotrNrre.l FoR
UNCoNFTNED
INTERFACE FLow
The
distances from sea level to
the
phreatic
surface
and
the
:nterface
are
represented
as
y
and 1", respectively
(see
Figure
I
),
both of
which vary with
position
in
the
horizontal
plane.
The
total
height
ofthe flow
region,
being the sum olr and
,,
:s
equal
to h; i.e.,
h=h1
*h"
(9)
,see
Figure
l).
By denoting
the
specific
masses
of
freshwater
-ind
salt
water
by
l, and
1",
respectively,
the
well-known
Ghy-
oen-Herzberg
formula
may
be
written with
this notation
as
h"=
hrVr/(1"-
lr))
(10)
'.see
Badon
Ghyben,
1888i Herzberg, l90ll. If
the
head
@
is
neasured
in
relation
to
some
impervious
base
that
lies a
dis-
:ance
ll,
below
sea
level,
one
may
write
(see
Figure I
)
g:fu*H,
(lt)
:hus
taking
into account that the
head
is
constant
along
a
:ertical
and is
equal
to
the
height of the
water
table.
The
:ollowing
relations
between
h
and
hl
and
between
h
and hu
are
:ound
from
(9)
and
(10):
h
:
hrll,/(l,
-
lr)l
:
h"(l"/lr)
(12)
Erpressing
r/ in terms ofd and
11,
by the
use
of(ll) and
:ubstituting
the result
in
(12),
one obtains
h
:
ll"/(t,
-
lr\l
-
v"/(1"
- lr))H,
(13)
This
equation becomes identical
with
the
general
relation
(3)
retween
and h if a
and
B
are chosen
as
a
:
l"/(1"
-
lr)
B
=
-ll"/(1"
-
lr)lH"
(14)
The
potential
for
unconfined interface
flow
is
obtained
lrom
5)
by
the use
of
(14)
for a
and
B,
which
yields
a
=
lkll"/(l"
-
lr)l@
-
H")z
+
Cui
(ts)
.,r'here
the
subscript ui
in
the
constant
C,r relers to
unconfined
:nterface
flow.
PorpNuel
FoR UNCoNFTNno
Flow
For
unconfined
flow the
well-known
relation
between
the
read
d
and
the thickness
ofthe
flow region, &,
is,
by
taking
the
rtmospheric
pressure
acting
at the
water
table as the zero
:eference and adopting the Dupuit-Forchheimer
assumption
'
cf.
Figure
I
),
gorl
ulace
This
equation again is a
special case of
(3)
in
which
a
and
0
now
are
one and zero,
respectively:
a:l 0:0
(17)
The
potential
,D
for unconfined flow is
obtained
from
the
general
equation
(5)
by
taking a
and
B
from
(17);
i.e.,
=
lkf,
+
Cu
(18)
The
index
u in
Cu refers
to
unconfined flow.
Exe.uplr
or
Mrxno
Uucounrurp-UwcoNntNrn
INrrnrncr
Flow
Cases
in which
both
unconfined
flow
and unconfined inter-
face
flow occur
in one
aquifer
can
now
be described by
the use
of
a single
potential
iD.
For
an
appropriate
choice
of
constants
C,;
and
C,
the
potential
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I 168
and
Stnecr: Coesrnl Aqurrrns
Z.Or,o
2
Lt4
ol
s|i|^ta$
fiw
Fig.
2. Interface flow in
a
shallow coastal aquifer
with
a
well.
l^pffica
,ffe+w
In
zone 2,
-
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Srucr:
Co.lsul Aeurrpns
elong the
tip
ol
the tongue,
while
the
harmonicity
of
iD
:hroughout
the
flow
region
follows from
the
continuity
equa-
:ion
(7)
and the
relation
between
the discharge
vector and the
:otential,
equation
(6).
The boundary
condition
along the coast
is that
the head
@
is
:qual to
the elevation
of sea
level
above
the
impervious
base
see
Figure
2),
-@
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Since
Q,
Q,o,
and
xp
are
positive
and since
(47)
must
be
:ulfilled,
p l= Q/(Q,ox.)l
must
be
restricted
to
values between
zero and
zr; i.e.,
0
-
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fi'|2
Srnncr:
Coesrer-
AeurFERs
plane
corresponding
to
the
tip of
the tongue
for four
different
In
zone
2 the
thickness
of the flow region
is constant and
situations,
of
which
three are unstable, are
represented
in
equal
to
H;
i.e.,
Figure
9
as curves
1,2,
3, and
4. The
corresponding
points
in
Figure
8
are
labeled
1,2,3,
and4, respectively.
Note
that
only
h
=
H
(54)
the
portions
ofthe
curves
in the
upper
halfplaney
>
0 are
ThepotentialiDinzone2isfoundfrom(3),(5),and(54)tobe
represented;
the complete
curves are
symmetric
in relation
to
thexaxis.
:kH*C"
(55)
The
boundary
between
the zones
is
at the
tip of
the tongue.
The head
at
the
vertical
surface
through
the tip ofthe
tongue
is
found from
(52)
by
setting
equal
to
.1,
a
procedure which
yields
Q:
H"(1,/11)
(56)
The
condition
that
(53)
and
(55)
for
be
equal at the
tip
ofthe
tongue,
wtrere
(56)
must hold, leads to
the
following
condition:
c"
-
c"t:
o'";
"
,'- kHH"lf
$7
llrtl
Il C"i
is chosen
to
be zero,
(57)
becomes
c'"i
=
o c":
**1"
-
lr
g,
-
kHH,l:
(5g
2
lr
"11
The
boundary condition along the coast
in
terms
of
lollows from
(52)
and
(53)
by setting equal to zero
(see
Figure
I
I
),
which
leads
to
=0
x=0
--
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Srucr:
Co,csreL
Aqulrrns
tt73
?tezo^.e,+io
)eel
ito
lowq
qviki.
l*n
'A5L
z
rcel
t
Q
=
Q.ox
+
fl
nlg;ffif',"
(60)
The
location
of the
tip of
the tongue
in
the x,
y
plane
is
lound
by setting
iD
equal
to
the
value
corresponding
to
:
H,(1,/l)
r
see
(56)).
This
value
is
found
from
either
(53)
or
(55),
with
the
aid ol
(58),
and
its substitution for
in
(60) yields
lt
ikH,,+
:
Q,ox
l1
,
Q
,-f
G-x,\'+Y'f"
;;tnl(x
-l
(6r)
It may
be
noted
that
the left-hand
side
of this expression
differs
lrom
that
of the
expression
obtained
for
the
case of
Figure
2,
equation
(36),
by
a
factor
l(H/H")'zlr/1").
The reader
nay
verify,
by
retracing
the
steps
taken
in the
stability
analysis
H
H5
zore
2
illustrated
in Figure
6,
that
interface
instability
occurs
if
t-(l-tr/n)'''
(62)
1+(1
-tr/o)'''
where
the subscript c
in
" refers
to
confined
flow
and
where
,
kH, l"-lr
a
:-
n":
U"u-=1-
u:
ffi
(63)
Equation
(62)
is represented
graphically
in Figure 8c,
as
is
seen
by
noticing that
Figure 8a
is
an
illustration o(47),
which
has
the
same
form as
(62).
Note,
however,
that now is to be
replaced by
"
(see
(63))
rather than by the expression
in-
dicated
in
the
graph
of
Figure
8.
It
may be
of
some
practical
interest
to
determine the loca-
tion
of the most inland
point
of the
tip
of the tongue. As is
seen
lrom
the
sketch
of
the location of the
tip
of the tongue
in
the x,
lLu
I
,,"rn
^":
,[,
-
4-.l"'+
[ln
L
7t)
7t
L-
Fil
"
kN,
r
us
'f
cl
hAL
r- t
Qror*
e{
:
Lr,Lo-
-'*-
tnle.rlco
lrowL
,4
q,
[=
0/(a;*1
2.O 2.5
Fig.
10.
Mixed confined-conflned
interface flow.
L"/x* and Lu/x*
as functions of the flow
parameters.
-
8/10/2019 Strack 1976 A Single-Potential Solution for Regional Interface Problems in Coastal Aquifers
10/10
tt'74
Srnlcr: ColstnL
Aqulrrns
y
plane given
in Figure
3, the
most
inland
point
ol
the
tip
of
the tongue is at
the
,r axis.
For
the mixed
uncon-
fined-unconfined interface flow
case
(cf.
Figure 2) the
ex-
pression
for
points
ofthe tip ofthe
tongue
is
given
by
(36).
The
maximum distance
from
the coast
to
the
tip
of
the tongue,
L,,
is iound by
substituting Lu for
x
and zero for
y
in
(36),
a
procedure
which
yields,
after
some modification,
(64)
with
and
p
lrom
(48).
A similar
expression is
obtained
for
the case
of
mixed
con-
fined-confined interface
flow
by the use of
(6
I
),
benefit
in
numerical methods such as the
finite
element
tech-
nique.
The
potential
iD
would
then replace
the
head
d
as the
dependent
variable
throughout the aquifer, the
necessity
tc
incorporate conditions along
the
boundaries between
th:
zones,
which
are
not known
a
priori,
thus being avoided.
Acknowledgments. The
research reported
in
this
paper
develope:
from work
done
at
the
Delft
University
of
Technology
and
ua.
supported by a
grant
from
the
Graduate
School of the
Universitl c'
Minnesota.
I
am
indebted
to
Steven
L. Crouch for
his
construclir.
comments.
RrpeneNces
Ackermann,
N. L.,
and
Y. Y.
Chang,
Salt
water interlace dun:.;
ground-water
pumping,
J.
Hydraul. Diu.
Amer. Soc.
Ciuil
Eng..
-
223-23t,197r.
Aravin,
V.
I.,
and
S.
N.
Numerov,
Teoriya Duizheniya
Zhidkoste:
Gazou
u
NedeJbrmiruemoi
Poristoi
Srede, Gosudarstvennoe
lzi.-
tel'stvo
Tekhniko-Teoreticheskoi
Literatury, Moscow,
1953.
Aravin,
V.
I.,
and
S.
N.
Numerov,
Theory of
Fluid Flow
in
Undefor""'
able
Porous
Media, Daniel
Davey,
New
York, 1965.
Badon
Ghyben.
w.,
Nota
in Verband
met
de
Voorgenomen
Putborr:;
Nabij Amsterdam,
Tijdschr. Kon. Inst.
Ing., 1888-1889,
8-22,
18:
Bear,
J.,
Dynamics
of
l-luids
in
Porous
Media,
Elsevier,
New
Yo:,.
t972.
Bear, J., and
G.
Dagan, Some exact solutions
of interface
problems :
means
of
the hodographic method,
J.
Geophys.
Res.,
64,
156:-
t572.
1964.
De
Josselin de Jong,
G.,
A
many
valued
hodograph
in an interf::.
problem,
Water
Resour. Res., 1(4), 543-555,
1965.
Dupuit, J., Etudes Thoretiques
et
Pratiques sur
le
Mouuement
a=
Eaux
dans
les Canaux Dcouuerts
et
Traers les
Terrains
Pennt:'
bles,
2nd ed.,
Dunod, Paris,
1863.
Forchheimer, P., Uber die Ergiebigkeit
von
Brunnen-Anlagen
u::
Sickerschlitzen,
Z.
Architekt.
Ing.
Ver. Hannooer,
32,
539-5t:
l 886.
Girinskii,
N. K.,
Le
potentiel complexe
d'un courant
surface
lib:=
dans
une
couche relativement mince
pour
k
=
flz),
Dokl. Akc:
Nau,t. S.SSR,
5
I
(5\,
341-342,
1946.
Cirinskii,
N.
K.,
Kompleksnyi
potentsial potoka
presnykh vod
..
slabo
naklonennymi
struikmi,
fiI'truyushchego
v vodopronitsaen
tolshche
morskikh
poberezhii,
Dokl.
Akad.
Noa&. ,S,S,SR, J8l:
559-56t,
t947
.
Henry,
H.
R.,
Salt
intrusion
into
freshwater aquifers,
"/.
Geophys.
Re'
64,
t9ll-19t9, t959.
Herzberg,
A.,
Die
Wasserversorgung einiger
Nordseebaden, Z.
Gc
'
beleucht.
Wasseruersorg., 44,
815-819,
824-844,
1901.
Strack,
O. D. L.,
Some cases
olinterface flow
towards
drains,
"I.
r.
M
ath.,
6,
175-191, 1972.
Strack,
O. D. L., Many-valuedness encountered
in
groundwater flcr
doctoral
thesis,
Dellt Univ. of
Technol., Delft,
Netherlands,
l9-:
(Received
November
10,
1975;
revised May
14, 1976
accepted June l, 1976.)
)\
:
2L+
4
ln
|
(+,/x,
-
t)'1"
xu
n
LQ,t*.+)
(65)
with
"
and
p
rom
(63).
The
distance
from
the
coast
to
the
most
inland
point
of
the
tip of
the tongue is represented as I"
in
(65).
Equations
(64)
and
(65)
are represented
graphically
in
Figure
ll.
It
may
be
noted
that
for
the
absence
of thewell
(Q
-
0),
(64)
and
(65)
reduce
to
the
knowh
expressions
lor
one-
dimensional
flow
[cf.
Bear,
1972, pp.
562 and
563, equations
(9.7.5)
and
(9.7.9)1.
OrHrn
AppLrcetloNs
The
technique
outlined
in
this
paper
is
applicable
to
the
general
class
ol inultiple-zone aquifer
problems,
with
the re-
striction that each
zone
must be either confined, unconfined,
unconfined
interface.
or confined interface flow.
Constant
rainfall
or
evaporation may
be
incorporated
into
problems
in
which
the upper
boundary
ofthe
flow region is a
free surface.
Depending
upon
the
complexity
ol
the
problem,
it
may
be
necessary
to
use
more
advanced techniques than the method
of
images used
in this
paper.
For
example,
one may
use con-
lormal mapping
techniques
incorporating
a
single
complex
potential
O,
defined
as
0=+i
where
is the
potential
defrned above. The stream
function
then is
defined
by the relations
fGirinskii,
1946)
Q,--
-a/Y Q':
+/ax
The
problem
can then
be solved
in
terms
oi
the
complex
potential,
similarly
to aquifer
problems
involving
only
one
type
ol
flow.
It
may be noted
that the technique is not restricted to
analytic
methods
of
solution but may
be
used with some
^"=
r*+
r
rn
laa#: "