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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

    1/10

    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

  • 8/10/2019 Strack 1976 A Single-Potential Solution for Regional Interface Problems in Coastal Aquifers

    2/10

    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:

    -

  • 8/10/2019 Strack 1976 A Single-Potential Solution for Regional Interface Problems in Coastal Aquifers

    3/10

    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,

  • 8/10/2019 Strack 1976 A Single-Potential Solution for Regional Interface Problems in Coastal Aquifers

    5/10

    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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    6/10

  • 8/10/2019 Strack 1976 A Single-Potential Solution for Regional Interface Problems in Coastal Aquifers

    7/10

    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

  • 8/10/2019 Strack 1976 A Single-Potential Solution for Regional Interface Problems in Coastal Aquifers

    8/10

    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

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    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#: "


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