WATER RESOURCES RESEARCH, VOL. 31, NO. 10, PAGES 2497-2502, OCTOBER 1995
Gravitational equilibrium moisture profiles in swelling soils
JagathC. Ekanayake
Manaaki Whenua-Landcare Research New Zealand Ltd., Christchurch,New Zealand
David
J. Painter
Departmentof Natural ResourcesEngineering,LincolnUniversity,Canterbury,New Zealand
Abstract. Extensionof the work of Philip (1969a, 1972) has shownthat the effectof
overburdenpressureon volume changeof swellingsoilsproducesequilibriummoisture
profilesentirelydifferentfrom thosepredictedwithout consideringthe overburden
pressureeffects.Volume changeof swellingsoilsaccompaniedby moisturechangesalso
affectsthe overburdenpressure.The total volume changeunder an overburdenpressure
occursin three differentshrinkagephasesfrom saturationto oven dry. Thesevolume
changecharacteristics
are usuallydescribedon the shrinkagesurface,a surfacedrawn as
void ratio e (volumeof voids/volume
of solids)versusmoistureratio O (volumeof water/
volumeof solids)for differentoverburdenpressures.
A relationshipfor the moisture
gradientis developedby assumingthat the overburdenpotential, a componentof the total
potentialof soil water, is a functionof moistureratio only acrossa small soil elementof a
long swellingsoil columnwhich consistsof large numbersof finite soil elements.The
moisturegradienthas complexbehaviorbasedon the propertiesof the three shrinkage
phaseson the shrinkagesurface.Distributionof equilibriummoisturepathsis then
explainedby evaluatingthe moisturegradientsat pointson an idealizedshrinkagesurface.
It is shownthat the soil at great depthscould either be saturatedor unsaturatedat
equilibrium.(Thesedepthscouldbe over 250 m for somesodiummontmorilloniteclays).
below shallowwater tables [Youngsand Towner, 1970]. Althoughthe load factor was correctedto include the effect of
The theoryof the hydrologyof swellingsoilis differentfrom overburdenpressureon the volume changeas given by (1)
that of rigid soil due to volumechangewith moisturecontent [Philip,1970;Groenevelt
andBolt, 1972],the equilibriummoisunderoverburdenpressure[Philip,1969a,b, c, 1972].In rigid ture distributionin swellingsoilshas not been correctedacsoil, moisture content increasestoward the water table where cordingly.
the soil becomes saturated. Below the water table, the satuThe load factor an is
rated moisturecontentremainsunchangedwith depthat equilibrium. The equilibrium moisture content in swellingsoils,
•,
(•)
however,couldevendecreasewith depthbelowthe watertable
due to volume changeunder the overburdensoil pressure.
undera constant
overburden
Such moisture profiles have been observedin the field by for thenth(i = n) soilelement
pressureP,; e is the void ratio, T is the absolutetemperature,
Talsma[1974].
Introducingan additionalpotentialcomponentto the total and O is the moisture ratio. Here "i" indexes soil elements of
potentialof soil water in rigid soil,Philip [1969a] described finite thicknessbeginningat the surhce (i = 0) and increases
three typesof equilibriummoistureprofilesin swellingsoils. with depth (Figure 1).
The work presentedin this paper examinesthe behaviorof
The new potentialcomponent,whichwas similarto that sugequilibrium
moistureprofilesin swellingsoil, consideringthe
gestedby ColemanandCroney[seePhilip,1969a],wascalledthe
volume
change
propertiesaffectedby both moistureratio and
"overburden
potential"12(O),definedby (de/dO)P,whereP is
overburden
pressure
[Ekanayake,1991]. Such equilibrium
the overburdenpressureandde/dO is the slopeof the shrinkagecurvee(O) for zero overburdenpressure.Usingthe char- moistureprofilesare describedusingan idealizedshrinkage
acteristicsof volumechangedeterminedby earlier experimen- diagrambased on the propertiesof the previouslyreported
tal data, Philip [1969a] suggestedthat all three types of experimentaldata of volumechange(seebelow).Final settlementsof civil engineeringstructurescouldbe predictedusing
moisture
profiles
reacha uniquemoisture
ratioOp (picnotatic
the relationshipe(O, P), if the equilibriummoistureprofileis
point),whichis alwaysunsaturatedat greatdepthsandwhere
evaluatedusingthe shrinkagecu•es obtainedfor a sufficient
the apparentwet relativedensityof the soil reachesits maxirange of overburdenpressures.
mum. The overburdenpressurewas assumedto have a negligible effecton the volumechange,and suchassumptions
cre- 2. Theo•
1.
Introduction
a(O,
P),=P]•
ated uncertainties
about the existence
of unsaturated
Copyright1995by the AmericanGeophysicalUnion.
Paper number 95WR01652.
0043-1397/95/95WR-01652505.00
zones
•-•
2.1. General Prope•ies of Swelling Soils
and the Shrin•ge Diagram
From earlier theoreticaland experimentalresultsthe following characteristicsof volume change of swellingsoil under
2497
2498
EKANAYAKE
AND
PAINTER:
MOISTURE
Po=OI-
ground surface
i=0
.,
i=n
Zn
I
PROFILES
IN SWELLING
SOILS
The shrinkagediagrame(19, P) (Figure 2) showsa collection of shrinkagecurvesei(19) for a seriesof soil elements
(Figure 1) beginningfrom an unloadedsoil surfaceas i =
O,..',n1, n,n + 1,...,m,...,m
+ 1,..., (m is
max, seebelow).
The void ratio of swellingsoilsdecreaseswith overburden
pressure[Chang and Warkentin,1968; Groeneveltand Bolt,
1972;Yongand Warkentin,1975;Talsma,1977;Giraldezet al.,
1983;Giraldezand Sposito,1983].During moistureremovalthe
void ratio at any given moistureratio reachesa minimum at a
largeoverburdenpressure,sayat Pma,,,(i -- m), andthisvoid
Pm
i=m
i=m+l
i=m+n
I
I
Pm+n
I
ratio is no longerreduciblewith further increaseof overburden
pressure;therefore the shrinkagecurves,em(©), em+ 1(©),
em+2(©), ''' , take a uniqueshapefor P > Pmax,(i > m).
Normal shrinkagephasesof all the shrinkagecurves,ei(t9)
for i = 0, 1, .-.,
n, n + 1, ...,
m, m + 1, ...,
create the
loadsurfaceon the e(19, P) diagram.On the load surface,soil
is saturated and e = 19, de/dP = d19/dP for P > 0 and
( Oe/O19)p
n = 1 [Groenevelt
and Bolt, 1972].Shrinkagecurves
have
a
unique
shape
for
P
-> Pmaxand (Oe/OP)o = 0, for
Figure 1. Idealized swellingsoil profile. Here "i" indexes
soil segmentsof finite thicknessbeginningat the surface(i = P -> Pmax'
Below the air entry points, (19 < 19a,) for all the soil
0) and increaseswith depthto minimumvoid ratio at i - m.
elements(i = 0, 1, ..., n, n + 1, ..., m, m + 1, ... ), the
e(19,P) surfacecreatesthe residualsurfacewhere(Oe/OP)o
< 0
overburdenpressureare consideredto be valid and applicable for 0 < P < Pmaxand (Oe/OP)o-- 0, for P _>Pmax,because
for swellingsoilswhich exhibit one-dimensionaland reversible shrinkagecurveshavea uniqueshapefor P -> Pmax'
volume changes[Tempany,1917;Haines, 1923;Lauritzenand
The load factora,, givenby (1), is relatedto the averageof
Stewart, 1941; Holmes, 1955; Warkentin and Bozozuk, 1961; (0e/019)o for overburdenpressures
Pi, i -- 0, 1, ..., n for
Fox, 1964;Philip and Smiles,1969;Philip, 1969a, 1971, 1972; 19> 19o.On the loadsurface,an = (Oe/O19)en
= 1 for 19 > 19a0
Groeneveltand Bolt, 1972;Sposito,1973; Talsma,1977;Giral- forP -->0. Thereforean < (Oe/O19)pn,
0 for 19a0> 19> 190'
dez et al., 1983;Giraldezand Sposito,1983].
And becauseof the compressibility
of swellingsoil,(0e/019)o
Volume changeof a swellingsoil under an overburdenpresand otn increasewith overburdenpressurefor 19a0> 19> 190'
sure takesplace in three shrinkagephasesfrom saturationto
19a0is the air entry value for zero overburdenpressure.
ovendry at a moistureratio equalto !9o. Volume reductionis
Sincethe void ratio is a nondecreasingfunction of the moisno longerpossiblefor moistureratios below 19o.Suchvolume
ture
ratio, (02e/O192)pn
> 0 in the residualphase[Philip,
changebehaviorcan be describedon a singleshrinkagecurve
1969a],
and
therefore
(00t/O19)p
n > 0 for 19a0> 19 > 19zn'
en(©) (sayfor thenthsoilelement,(i = n)), obtained
for a
And
also
because
(Oe/O19)en
=
1
in
the normalshrinkage
phase
constantoverburdenpressurePn as normal,residual,and zero
shrinkagephases.
The normal shrinkagephaseis associatedwith the saturated
condition,where void ratio is equal to the moistureratio and
changeof moistureandvoidvolumesare equal.Thereforee =
!9 and (Oe/O19)p
n = 1 for !9 _>•}an where•}an is the air entry
point under the overburdenpressurePn.
The residual shrinkagephase occursduring further drying,
after air beginsenteringthe systemat the air entry point !9an
and before the shrinkagelimit ©zn' In the residualshrinkage
phase,voidratio is largerthanthe moistureratio,andthe change
of voidvolumeis lessthanthe changeof moisturevolume.Therefore e > 19and (Oe/O19)en
< 1 for 19an) © ) 19zn'
The zero shrinkagephaseoccursafter the shrinkagelimit
19•n,where the final void ratio under the overburdenpressure
Pn remainsunchangedat e0nwith further removalof moisture
and therefore (Oe/O19)p
n = 0 for Ozn ) 0 ) 00.
These characteristicsare valid for overburdenpressures,
Pi = Po, P•, '", Pn, Pn+i, ''', Pm, Prn+l, ''', (i -- O,
1,...,n,n
+ 1,...,m,m
+ 1,..',P--PmaxfOri
=m).
Consolidationstudiesof Wyoming bentonitewith 90-95%
and(Oe/O19)en
= 0 in thezeroshrinkage
phase,
(02e/0192)e
n= 0
and(Oan/O19)en
= 0 withinthezeroandnormalshrinkage
phases.
The total potentialof soil water •r
acrossthe t/th soil
element (i = n), with a finite thicknessunder a constant
overburdenpressurePn (headequivalent),variesaccordingto
=
- (z.. + z) + ...(o)
?.. +
.,/..(o) arz
(2)
where(Z n -{-z) is the gravitationalpotential(headequivalent,
vertical ordinate taken positivedownwardfrom the top soil
surfaceto the top of thet/thsoilelementandz is measured
fromthetopsurface
of thenth(i -" n), soilelement).
HereOt
n
is the load factorgivenby (1), and •/n(©) is the apparentwet
relativedensity(wetbulk densityof soil/density
of water).Both
otn and •/n are assumedto be uniquefunctionsof 19acrossthe
finite soil elementunder the constantoverburdenpressurePn.
ß (19)is the matricpotentialof the unloadedsoil (headequivalent) and is a unique function of the moistureratio [Philip,
montmorillonite
show that the final irreducible
void ratio
1969a;Towner,1981].(Note that otn and •/n are determinedfor
(eorn)undersaturatedconditionscouldreachbeyond44 bars overburdenpressurePn).
The moisture
gradientacross
thet/thsoilelementat equiof pressure[Mesri and Olson, 1971]. This is approximately
equal to a soil depth of 250 m, with no surfaceloads.
librium (•r constant)canbe obtainedfrom (2) as
EKANAYAKE
AND
PAINTER:
MOISTURE
PROFILES
IN SWELLING
SOILS
2499
normal
phase
zero
phase
for
P=Pn
resictual
phase
for
P=Pn
(,3e/jO)e•l
I
I
'
'
•
, ,
"• eon)
l,Ozn'ezn)
•--•0,
. _.,.,
>
/ innormal
phase
P_O
0.,(.Oan,ean}
/
' Pm,•Pm P. P.-•/
,
I
constant
P/•/•//•.-J
/
.--,/
i
•/'•//'••J•
•
loadsurface
(Be/SO)p=1
in[ •Ot r' n
-'rentry
po
J
,,/-
,/,-
/
J
J
,.,"
P• P•
/•
e(O)
curve
for
P=P0=O
•
se norma
phas
base
=o
zerop
c•
(• ----0
=0
e.
=e atP e
m•n Om m, •
• '•min•,,otca,,t •
•
jl:)
I'
•
_
"0
-
•
,•
•I:•
I
I
I
p_••-""'•-•.•
I
verl::)•rclen
Pressure
(p)
, •
•
emax
P ,8
oo
Figure 2. Idealizedshrinkagediagrame(©, P). ©,n, e,n is the moistureratio andvoid ratio at air entry
pointfor Pn; ©zn,eznis the shrinkage
limit for Pn; andemin is the irreducible
voidratioat ©ofor Pmax.
dO
--
[l -- an( O) ')ln
( O) ]
=
dT
•
Tn(19)=
dan(O)
+ dO Pn+
(19 q- ')Is)
(4)
n
Zn
7n(O)
dz
where
7sistherelative
densiW
ofsolids.
Here 7, (O) hasonlyone m•imum givenby 7•, (O•,) =
1
1/(Oe/OO)•n,at O•n for O•n < 0 < O,n [Phil½,1969a].
=Mn(O)
(3) Because
thevoid
ratio
decreases
with
overburden
pressure,
7, (O) increases
with overburdenpressureat anygivenmois-
ture ratio. Similarlyto the behaviorof the en(19) curve,the
7,•(19) curveapproachesthe '•/m(19) curveas P approaches
fZn+!
of Tm(19) remain
At largedepths
thetermJZn T(19)dz is a smallquantity Pm. Thereafter the generalcharacteristics
compared
to largeoverburden
pressures
andmaybe ignoredin unchangedwith P (Figures3a and 3b).
calculating
Mn(19).
Approximating
(Pn+•-- Pn)byJ'zZn
n+•
7(19)
dz The load factora n(19) givenby (1) is alwayslessthan unity
andusingdP/d19= %, dz/d19across
the/,/thsoilelement, for 19< 19a0but an(19) -• 1 as19-• 19aO.It is alsoclearfrom
properties
of volumechange
described
equilibriummoistureprofilesare simplytracedon the e = 0 (1) and'thegeneral
planeby evaluating
dz/d19= Mn(19) acrosseachsoilelement, abovethat an(19) = 0 for 19•,, > 19 > 19o(zero shrinkage
andan(19)< 1 for19zn
< 19< 19an
(residual
shrinkbeginningfrom the surfacesoil elementi = 0 at equilibrium phase)
with a knownsurfacemoistureratio 19Po,and surfaceover- agephase).Therefore,similarlyto the behaviorof the Tn(19)
the a m( 19) curveas
burden
pressure
Poandcontinuing
i = 1, ..., n, n + 1, .:.., anden(19) curves,an(19) alsoapproaches
m, rn + 1, ... toward large depths where P > Pm' To p _• pm where its shaperemainsunchangedfor P >- Pm'
achievethis,the behaviorof Mn (19) mustbe well understood. From (3) at a: certain moisture ratio, say
The functionsT.(19), a.(19), dT/d19, anda.T.(19) will now Tn(19/n)an(19/n)= 1 and Mn (19/,,) -• +_oo.It has been
be examinedseparatelyand then together.
shownthat an < (Oe/O19)pn,o
for 19"o> 19> 19oandboth
The apparentwet relativedensityTn(t9), whichis givenby functionsincreasefrom 0 at 19oto 1 as 19 -• 19a0.Therefore
= 1 at 19dnmust be satisfied
(4), is determinedfor a swellingsoilunderoverburden
pres- the condition(Oe/O19)pn19n
surePn- It is alwaysgreaterthan unity for 19 > 19obut ap- beforethe conditiona n'Yn= 1 is satisfiedat 19,rnin 19 < 19
the function anTn(19) approaches
proachesunity as 19-• 19max,
where 19max
is a large quantity And also as 19 -• 19max,
greaterthan the saturatedmoistureratio withoutoverburden unity asboth Tn(19) and a n(19) approach1 at largemoisture
pressure--19max
>>> 19a0'Figure3 showsthe idealizedfunc- ratios.Therefore19,rnliesbetweenthe air entrypoint 19aOand
tions:en(19), 7n(19), andan7n(19) for soilelementsi = 0, n, maximumapparentwet relativedensitypoint 19dn(Figure3c).
and m.
For all swellingsoilswe takedT/d© > 0 for 19max
> 19>--190
where Z,+ • - Zn = AZ is the thicknessof the small soil
element.
,;
2500
EKANAYAKE
PAINTER:
zerophase
Can
Po=O
ezo_/
eon
Pn
._o Coo
earn
_
_
eom
I
I
I
eo=O
eO
•'o
eam
ean
eao
•
PROFILES
It hasalready
beenshown
thatOa0> O/n > Oan.It isalso
the residual surface.
'Ydn
7=1
SOILS
importantto knowthe exactlocationof the point O/n to trace
the equilibriummoistureprofilesbecauseO/n couldbe either
on the lead surface(O/n > Oan) or on the residualsurface
(0in < Oan) (note that Odn< Oan < OaO). It is certainthat
OIn (at which dO/dP = 0) cannotlie on the lead surface
becauseon an ideallead surface,soilis saturated(e = O) and
an increaseof overburdenpressurealwaysreducesan equal
amountof moistureratio and void ratio (de/dP = dO/dP •:
0 for P -> 0). But, in the residualshrinkagephase,overburden
(a)
pressurecanbe increasedwithoutchangingthe moistureratio,
whilechanging
thevoidratio (00/OP)e = 0 and(Oe/OP)o•:0
is onlypossiblein the residualphasewhereO < Oan). Thereemax fore Oin mustlie within the rangeOan < OIn < Oan and on
'Ydm
•.• 7dO
IN SWELLING
2.3. The Location of Oln
residual
phase
• normal
phase
Po=O Po=O
;/ Po
=0
eao
•
MOISTURE
I
em
•
AND
•
•
•
3. Equilibrium Moisture Profiles
in Swelling Soils
Po
=0
edmeam
edn eanedO eao
•
Pm
ctn7n=l
Usingthe similaritybetweendz/dO anddP/dO, the behavior of thesemoistureprofilesis mostconvenientlydescribedon
ema
x the e = 0 plane usingthe propertiesof the functionMn(O)
describedabove, as shownin Figures4b, 5b, and 6b. Three
differentequilibriumconditionsare describedaccordingto the
surfacemoistureratio O•,o and surfacezero overburdenpressure Po = 0. The same method is applicablefor any other
surface moisture
ratios and surface loads.
(c)
elmednelnean elO
•
emax
moisture
ratio
Figure
3.Idealized
functions
of
(a)
e(O),
(b)
7(O),
and
(c)
aT(O)
for
overburden
pressures,
Po
=0,
Pn,
and
Pmax'
mal,
residual,
andzero
shrinkage
phases
are
shown
forPoNor=0
inFigure
3a.
•'
Pmax
.....
a2
a3I
,
•,
,
saturabng
point,
[Philip,
1969a;
Towner,
1981].
Equation
(3)
describes
the
behavior
ofthe
moisture
gradient
dz/dO
=Mn(O
).Mn(O
) >0forO< Oin
and
approaches
+ *:asO-• Oin.
Mn(O
) < 0forO> Oin
and
Mn(O) approaches
- *: asO approaches
O/nfromO,•n;alsonote
that(OMn(O)/OO)p
n > 0 forO > O/n.
Pr
-
(a)
2.2. The Functionse(Oa,Pa) and e(Ox,Px)
eao
All shrinkagecurvesfor soil elementsi = 0, 1, 2, ..-,
n, ..., m, ..., rn + 1, ... have air entry pointsat the mois(b)
ture ratios O,o, O,•, O,2, '", O,,,, ..., O,•m, O,•m+• and
alsoM,(O) = dz/dO • dP/dO ---->+_• at moistureratios
r \
O/o, O/•, /2,'",
O/n,''',
O/m,''',
O/m+•. All these
points form two curvese(O,, P,) and e(O/, P/) on the
shrinkagesurface.Becausethe moistureratios Oan and
decreasewith the overburdenpressureuntil P - Pm•, these
two curvesare concaveto the P axis,and sincethe shrinkage
saturating
point iI
a3 •
• /a 2 (eas,
curveshavea uniqueshapefor P -> Pm•.,,theybecomeparallel
Zs)
iI
to the P axisfor P -> Pm•.,-These two curvescan be drawnon
the e = 0 plane,asP(O,•) andP(O/) (Figures4a, 5a, and6a). It
Zmax
mustbe noted that accordingto the propertiesof Mn(O) describedin the previousparagraph,dO/dP[=(OO/OP)e=O]
> 0 Figure 4. Idealizedequilibriummoistureprofilesfor surface
for P(O) < P(O/), dO/dP< 0 for P(O) > P(O/), andon P(O/), moistureratio, O•,o < O/o and Po - 0. Curvesa3 according
dO/dP • dO/dz = O.
to Philip [1969a].
I I
EKANAYAKE
AND
PAINTER:
MOISTURE
PROFILES
IN SWELLING
SOILS
2501
3.1. Case a: When the Surface Moisture Ratio Opo Lies
in O•o > Opo > 0 and Po=0
This is an unloadedsurfacewherethe location((9eo,Po = 0)
is shownon the idealizeddiagramsof P versus(9 andZ versus
in Figures4a and 4b. For (9/0> (9/0,Mo((9eo)> 0, dz/d(9> O,
and dP/d(9> 0 on the groundsurfacewith no load (Po = 0).
Therefore(9 increases
withZ andP towardtheP((9/) boundaryas
long as P((9) < P((9/). It meetsP((9/), sayat (9/r as shownin
Figure4a. The point(9/rlieson oneof the shrinkage
curves,say
er((9),whichisdetermined
for theoverburden
pressure
Pr (equivalentdepthZr). At the point((9/,•Pr),Mr(©/r) '-->+% dP/d(9
+% anddz/d(9--> +•. Thereforethe moisturepath crossing
the
P((9/) boundarywill alwaysbecomeparallelto the P andZ axes
as shownin Figures4a and 4b. When P((9) > P((9/), M(©) =
dz/d(9 • dP/d(9 < 0, and thereforethe moisturepath continues
towardthe P axisbut awayfrom the P((9/), becauseon
dP/d(9--> +•. Further continuationof the moisturepath could
eitherintersecttheP((9a)boundary,
sayat (9as(on the shrinkage
curve,es((9)),whichis determined
for P•, (equivalent
depthZ
or crossthe shrinkage
curveem((9
) for P = Pmax(Zm) at a point
I
(9*min the range(9am• ©*m • (9Imasshownby al in Figure4a.
The moisturepathwill not crosstheP((9/) boundaryfor P > Pr
because
onP((9/), dP/d(9--> +c• anddz/d(9--> +c•. On the other
hand,if it intersects
P((9a)at (9a•,whichisontheloadsurface,
the
soilbecomes
saturated.
Thereafterthemoisturepathisforcedto
remainon the load surfacefor P -> P•, becauseon the load
i
Cl
: /'
I , /
/
iC/3
/
.
.',
saturatingpoint
,,
I
'/
'/
',
I
'/
'/
',
Zma
x ........
surface,e = (9, and further increasesin P and Z will result in a
/
C2
(b)
similarreductionof bothe and (9, forcingthe soilto remain Figure 6. Idealized equilibriummoistureprofilesfor surface
saturatedand on the load surfacewith increasingdepth.As the moistureratio, (9po,(9aO> (9/,0 > (9/0 andPo = 0. Curves
moisturegradientis stillnegativein P((9) > P((9/), the moisture c3 accordingto Philip [1969a].
path approaches
the P((9a) boundaryonce again,for P > Ps,
whereit remainssaturatedat ©amthroughoutthe depthfor P >
Pmaxon the load surfaceas shownin Figures4a and 4b.
If the path reachesa point (9, m where(9Im ( © * m ( © am,
as shownby al in Figure 4a, due to the negative moisture
gradient for P((9) > P((9/) the moisture path continues
towardthe P(O/) boundary,whichis parallelto the P axisfor
P > Pmaxat (9/m, where it remainsunsaturatedthroughout
the entire depth.Philip's[1969a]moistureprofilesunder similar conditionsare shown by a3 in Figures 4a and 4b. If a
surfaceload Pt is present on the surface,the moisture path
beginsat a point Po = Pt and (9 = (9or.
Pmax
saturating
point',
Ps
__
_
i
_
, P(•a)(a)
OIm
Oam
•)as
I
0 0=OP0 Oa0
I
(b)
Zs
This point is locatedon the (9 axisas shownin Figures5a
and5b.At (9/0 wherethe boundaryP((9/) begins,Mo((9/o) =
dz/d© = 4_-_
•. Soonafter the moisturepath beginsparallel to
the P axis at (9/o, it enters the region P((9•) < P((9) <
P((9a) where the moisturegradientbecomesnegative.Thereafter, the behavior is identical to that in case a above. As (9
_
b2
Zma
x - _
3.2. Caseb:O•,o= O,o,andPo= O'
point
• b3
decreases
with depth,it reacheseither the P((9a) boundaryin
P < Pmaxor the P((9D boundaryin P > Pmax'If it intersects
P((9a) in P < Pmaxat (9aswhereit becomessaturated,it will
remain saturatedon the load surfacethroughoutthe depth
until it reaches(9amonceagainfor P > Ps ,(pathb2), where
it remainssaturated.If it doesnot intersectP((9a) for P <
Pmax,it will be unsaturatedthroughoutthe entire depthand
(9ImforP > Pr,where
it remains
unsaturated
(path
Figure 5. Idealizedequilibriummoistureprofilesfor surface willreach
moistureratio, (9/,0 = (9/0 and Po = 0. Curvesb3 according bl). The equilibriummoistureprofilesof Philip [1969a]under
similarcondition
s are shownby b3 in Figures5a and5b.
to Philip [1969a].
2502
EKANAYAKE
AND
PAINTER:
MOISTURE
,.
3.3. Case c: O,,o > Ot.o > O•roand Po = 0
PROFILES
IN SWELLING
SOILS
Giraldez,J. V., and G. Sposito,A generalsoilvolumechangeequation,
II, Effect of load pressure,Soil Sci. Soc.Am. J., 47, 422-425, 1983.
In this casethe moisturegradient is negativefrom the be- Giraldez,J. V., G. Sposi•t
9, and C. Delgado,A generalsoilvolume
ginning at Po = 0. Therefore the moisturepath reaches
changeequation;I., The two-parametermodel,SoilSci.Soc.Am. J.,
47, 419-422, 1983.
P(Oa) at {9as,where it becomessaturated and will remain
saturatedthroughout
the entiredepth(Figure6) andreach
P(Oa) onceagainat depthsP > Ps (path c2). The moisture
ratio at depthswhereP -> Pmaxmaybecomesaturated(c2) or
unsaturated(cl) dependingon whethe•rthe moisturepath intersects
P(Oa) or not,asin casesa andb. If the surfacesoilis
saturated
whereOeo > 19a0,the entiremoisturepathlieson
theloadsurfac
e untilit reaches
!9amforP > Ps,wherethesoil
remains
saturated.
Groenevelt, P. H., and G. H. Bolt, Water retention in soil, Soil Sci.,
113(4), 238-245, 1972.
Haines,W. B., The volumecha,
ngeassociated
withvariationsof water
• content in soil,J. Agric. Res., 13, 296-310, 1923.
Holmes,
J.W.,Watersorption
andswelling
of clayblocks,
J.SoilSci.,
6(2), 200-207, 1955.
Lauritzen, C. W., and A. J. Stewart,Soil volume changesand accompanyingmoistureand pore-spacerelationships,
Soil Sci. Soc.Am.
Proc., 6, 113-116, 1941.
Mesri, G., and R. E. Olson, Consolidation characteristicsof montmo-
rillonite,Geotechnique,
21(4), 341-352,1971.
Equilibrium
moisture
profiles
for anyotherequilibrium
conPhilip,
J.
R.,
Moisture
equilibrium
in the verticalin swellingsoils,1,
ditionscanbe tracgdusingthe sameprocedureandthe given
Basictheory,Aust.J. Soil Res., 7, 99-120, 1969a.
Philip, J. R., Moisture equilibriumin the verticalin swellingsoils,II,
Applications,Aust.J. SoilRes.,7, 121-141, 1969b.
Philip, J. R., Hydrostaticsand hydrodynamics
in swellingsoils,Water
4. Conclusion
Resour.Res.,5(5), 1070-1077, 1969c.
The equilibriummoistureprofilesin swellingsoilhavebeen Philip,J. R., Reply,WaterResour.Res.,6(4), 1248-1251,1970.
shownto be differentfrom thosepredictedby earlier authors Philip, J. R., Hydrologyof swellingsoils,in Salinityand Water Use,
ProceedingSymposium
Australia, edited by T. Talsma and J. R.
surface
conditions.
due to the effectsof overburden
pressure
on the volume
change.Soil at great depthsmay be either saturatedor unsat-
Philip,pp.125-139,Macmillan,
NewYork,1971.
Philip, J. R., Recentprogressin the theoryof irrigationand drainage
of swellingsoils,paper presentedat the 8th Congressof the Interurated.It is clearthat if the soil becomessaturated,it will
remain
saturated
at19am
atdepths
where
P > Pmax
asshown nationalCommissionon Irrigation and Drainage,Int. Comm. on
Irrig. andDrain.,Varna,Bulgaria,1972.
byprofile
a2in l•igures
4aarid
•4b.HOwever,
soilatgreat Philip,
J. R., •andD. E. Smiles,Kineticsof sorptionandvolumechange
•epths
could
beunsaturated
andremain
Unsaturated
atamois-
in three componentsystems,
Aust. J. Soil Res., 7, 1-19, 1969.
t•ur•e
ratioO•rm
for1.arge
depths
(P > Pmax)
asshown
byprofile Sposito,G., Volumechangein swellingclays,SoilSci.,115(4),315-320,
al in Figures4a and 4b. However,only two equilibriummois-
ture ratiosexistat great depth in swellingsoils.Th9se two
moisture
ratios,Ottoand{9am,areverysmall,astheyare
dete.rmined
forPmaxMoisture
profiles
underequilibrium
conditionscanbe determined
for a givensurfacemoistur
e ratio
andthe loading,if the detailsof the shrinkage
surfaceare
kndwn.Theshrinkage
surface
canbedetermined
froma series
1973.
Talsma,T., Moisture profilesin swellingsoils,Aust.J. Soil Res.,12,
71-75, 1974.
Talsma, T., A note on shrinkagebehaviourof a clay paste under
variousloads,Aust.J. Soil Res.,15, 275-277, 1977.
Tempany,H. A., The shrinkageof s0ils,J.Agric.Res.,8, 312-333,1917.
Towner, G. D., The correction of in situ tensiometer readings for
overburdenpressures
in swellingsoils,J. Soil Sci.,32(4), 499-504,
1981.
of shrinkage
curvesunderone-dimensional
volumechange Warkentin,B. P., and M. Bozozuk,Shrinkingand swellingproperties
obtainedfor differentoverburden
pressures
fromP = 0 to the
overburden
pressure
relevant
totherequired
depth.
Once
the
equilibrium
moisture
profileis determined
fora soilwitha
of•o Canadian
clays,
Proc.
Int.Conf.SoilMech.
Found.
Eng.,1,5,
851-855, 1961.
Yong, R. N., and B. P. Warkentin, Soil Propertiesand Behaviour,
Elsevier Sci., New York, 1975.
surfaceload (e.g., a civil engineeringstructure),an e(O, P) Youngs, E.G., and G. D. Towner, Commentson "Hydrostaticsand
relationshipfor the entire depth is known. This relationship
hydrodynamics
in swellingsoils"by J. R. Philip,WaterResourc.
Res.,
6(4), 1246-1247, 1970.
canthenbe usedto evaluatethe void ratio changealongthe
entire depth (i.e., the total settlement)at gravitationalmoisJ. C. Ekanayake,Manaaki Whenua-LandcareResearchNew Zealture equilibrium.
References
Chang,R- K., and B. P. Warkentin,Volume changeof compactedclay
soil aggregates,
Soil Sci.,105(2), 106-111, 1968.
Ekanayake,J. C., Soil water movementthroughswel!ingsoils,Ph.D.
thesis,Lincoln Univ., Canterbury,New Zealand, 1991.
Fox, W. E., A studyof bulk densityand water in a swellingsoil,Soil
Sci.,98(1), 307-316, 1964.
and Ltd., P.O. Box 69, Canterb.
ury Agricultureand ScienceCentre,
GeraldStreet,Lincoln,NewZealand.(e-mail:ekanayaj@landcare.cri.nz)
D. J. Painter,Departmentof Natural ResourcesEngineering,P.O.
Box84,LincolnUniversity,Canterbury,NewZealand.(e-mail:painter@
lincoln.ac.nz)
(ReceivedMay 9, 1995;acceptedMay 24, 1995.)